Continuity and Limits
Disclaimer: These are my personal notes compiled for my own reference and learning. They may contain errors, incomplete information, or personal interpretations. While I strive for accuracy, these notes are not peer-reviewed and should not be considered authoritative sources. Please consult official textbooks, research papers, or other reliable sources for academic or professional purposes.
1. Limits of Functions
1.1 Intuitive Definition
The limit of $f(x)$ as $x$ approaches $a$ is $L$ if $f(x)$ gets arbitrarily close to $L$ as $x$ gets arbitrarily close to $a$.
1.2 Precise ($\epsilon$-$\delta$) Definition
Definition: $\lim_{x \to a} f(x) = L$ if:
$$\forall \epsilon > 0, \exists \delta > 0 \text{ such that } 0 < |x - a| < \delta \Rightarrow |f(x) - L| < \epsilon$$
This definition excludes the point $x = a$ itself.
1.3 One-sided Limits
Right-hand limit: $\lim_{x \to a^+} f(x) = L$ if:
$$\forall \epsilon > 0, \exists \delta > 0 \text{ such that } 0 < x - a < \delta \Rightarrow |f(x) - L| < \epsilon$$
Left-hand limit: $\lim_{x \to a^-} f(x) = L$ if:
$$\forall \epsilon > 0, \exists \delta > 0 \text{ such that } 0 < a - x < \delta \Rightarrow |f(x) - L| < \epsilon$$
Theorem: $\lim_{x \to a} f(x) = L$ if and only if $\lim_{x \to a^+} f(x) = \lim_{x \to a^-} f(x) = L$.
1.4 Infinite Limits
$\lim_{x \to a} f(x) = +\infty$ if:
$$\forall M > 0, \exists \delta > 0 \text{ such that } 0 < |x - a| < \delta \Rightarrow f(x) > M$$
1.5 Limits at Infinity
$\lim_{x \to \infty} f(x) = L$ if:
$$\forall \epsilon > 0, \exists N > 0 \text{ such that } x > N \Rightarrow |f(x) - L| < \epsilon$$
2. Properties of Limits
2.1 Limit Laws
If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$, then:
- Sum: $\lim_{x \to a} [f(x) + g(x)] = L + M$
- Difference: $\lim_{x \to a} [f(x) - g(x)] = L - M$
- Product: $\lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M$
- Quotient: $\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}$ (if $M \neq 0$)
- Power: $\lim_{x \to a} [f(x)]^n = L^n$
- Root: $\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{L}$ (if $L > 0$ for even $n$)
2.2 Squeeze Theorem
If $g(x) \leq f(x) \leq h(x)$ near $a$ and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.
2.3 Composition of Functions
If $\lim_{x \to a} g(x) = b$ and $f$ is continuous at $b$, then:
$$\lim_{x \to a} f(g(x)) = f(b) = f\left(\lim_{x \to a} g(x)\right)$$
3. Continuity
3.1 Definition of Continuity
A function $f$ is continuous at $a$ if:
$$\lim_{x \to a} f(x) = f(a)$$
This requires three conditions:
- $f(a)$ is defined
- $\lim_{x \to a} f(x)$ exists
- $\lim_{x \to a} f(x) = f(a)$
3.2 $\epsilon$-$\delta$ Definition of Continuity
$f$ is continuous at $a$ if:
$$\forall \epsilon > 0, \exists \delta > 0 \text{ such that } |x - a| < \delta \Rightarrow |f(x) - f(a)| < \epsilon$$
3.3 Types of Discontinuities
- Removable discontinuity: $\lim_{x \to a} f(x)$ exists but $f(a)$ is undefined or $f(a) \neq \lim_{x \to a} f(x)$
- Jump discontinuity: Left and right limits exist but are different
- Essential discontinuity: At least one one-sided limit doesn't exist
3.4 Continuity on Intervals
- Continuous on $(a,b)$: Continuous at every point in the interval
- Continuous on $[a,b]$: Continuous on $(a,b)$, right-continuous at $a$, left-continuous at $b$
4. Properties of Continuous Functions
4.1 Algebra of Continuous Functions
If $f$ and $g$ are continuous at $a$, then so are:
- $f + g$, $f - g$, $f \cdot g$
- $\frac{f}{g}$ (if $g(a) \neq 0$)
- $|f|$, $f^n$ (for positive integer $n$)
4.2 Composition of Continuous Functions
If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then $f \circ g$ is continuous at $a$.
4.3 Continuity of Elementary Functions
- Polynomials: Continuous everywhere
- Rational functions: Continuous except where denominator is zero
- Trigonometric functions: $\sin x$, $\cos x$ continuous everywhere; $\tan x$ continuous except at odd multiples of $\frac{\pi}{2}$
- Exponential and logarithmic: $e^x$ continuous everywhere; $\ln x$ continuous for $x > 0$
- Inverse trigonometric: Continuous on their domains
5. Important Theorems
5.1 Intermediate Value Theorem
If $f$ is continuous on $[a,b]$ and $k$ is between $f(a)$ and $f(b)$, then there exists $c \in (a,b)$ such that $f(c) = k$.
Corollary: If $f$ is continuous on $[a,b]$ and $f(a)$ and $f(b)$ have opposite signs, then there exists $c \in (a,b)$ such that $f(c) = 0$.
5.2 Extreme Value Theorem
If $f$ is continuous on a closed and bounded interval $[a,b]$, then $f$ attains its maximum and minimum values on $[a,b]$.
5.3 Uniform Continuity
Definition: $f$ is uniformly continuous on $D$ if:
$$\forall \epsilon > 0, \exists \delta > 0 \text{ such that } \forall x,y \in D: |x-y| < \delta \Rightarrow |f(x)-f(y)| < \epsilon$$
Theorem: If $f$ is continuous on $[a,b]$, then $f$ is uniformly continuous on $[a,b]$.
6. L'Hôpital's Rule
If $\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0$ or $\pm\infty$, and $\lim_{x \to a} \frac{f'(x)}{g'(x)}$ exists, then:
$$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$$
6.1 Indeterminate Forms
- $\frac{0}{0}$, $\frac{\infty}{\infty}$
- $0 \cdot \infty$ (rewrite as $\frac{0}{1/\infty}$ or $\frac{\infty}{1/0}$)
- $\infty - \infty$ (factor or rationalize)
- $0^0$, $1^\infty$, $\infty^0$ (use logarithms)
7. Limits and Continuity in Metric Spaces
7.1 Metric Space Definition
A metric space $(X,d)$ consists of a set $X$ and a distance function $d: X \times X \to \mathbb{R}$ satisfying:
- $d(x,y) \geq 0$ with equality iff $x = y$
- $d(x,y) = d(y,x)$ (symmetry)
- $d(x,z) \leq d(x,y) + d(y,z)$ (triangle inequality)
7.2 Convergence in Metric Spaces
A sequence $(x_n)$ converges to $x$ if:
$$\lim_{n \to \infty} d(x_n, x) = 0$$
7.3 Continuity in Metric Spaces
$f: (X,d_X) \to (Y,d_Y)$ is continuous at $a$ if:
$$\forall \epsilon > 0, \exists \delta > 0: d_X(x,a) < \delta \Rightarrow d_Y(f(x),f(a)) < \epsilon$$
8. Code Examples
# Python implementations for limits and continuity analysis
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import fsolve
import sympy as sp
class LimitAnalyzer:
def __init__(self, func, symbolic_func=None):
"""
Initialize with a function
func: numerical function (lambda or def)
symbolic_func: symbolic expression for exact computation
"""
self.func = func
self.symbolic_func = symbolic_func
def numerical_limit(self, a, side='both', h_values=None):
"""
Compute limit numerically by approaching from sides
side: 'left', 'right', or 'both'
"""
if h_values is None:
h_values = [10**(-i) for i in range(1, 16)]
results = {'left': [], 'right': [], 'h_values': h_values}
for h in h_values:
if side in ['left', 'both']:
try:
val = self.func(a - h)
results['left'].append(val)
except:
results['left'].append(np.nan)
if side in ['right', 'both']:
try:
val = self.func(a + h)
results['right'].append(val)
except:
results['right'].append(np.nan)
return results
def plot_limit_behavior(self, a, interval=0.1, num_points=1000):
"""Plot function behavior near point a"""
x_left = np.linspace(a - interval, a, num_points//2, endpoint=False)
x_right = np.linspace(a, a + interval, num_points//2, endpoint=False)[1:]
try:
y_left = [self.func(x) for x in x_left]
y_right = [self.func(x) for x in x_right]
except:
# Handle functions that might have domain restrictions
y_left, y_right = [], []
for x in x_left:
try:
y_left.append(self.func(x))
except:
y_left.append(np.nan)
for x in x_right:
try:
y_right.append(self.func(x))
except:
y_right.append(np.nan)
plt.figure(figsize=(10, 6))
plt.plot(x_left, y_left, 'b-', label='Left approach', linewidth=2)
plt.plot(x_right, y_right, 'r-', label='Right approach', linewidth=2)
plt.axvline(x=a, color='k', linestyle='--', alpha=0.7, label=f'x = {a}')
# Try to plot the function value at a
try:
fa = self.func(a)
plt.plot(a, fa, 'go', markersize=8, label=f'f({a}) = {fa:.3f}')
except:
plt.plot(a, 0, 'ro', markersize=8, label=f'f({a}) undefined')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.title(f'Function behavior near x = {a}')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()
def symbolic_limit(self, a, side='both'):
"""Compute limit symbolically using SymPy"""
if self.symbolic_func is None:
return "Symbolic function not provided"
x = sp.Symbol('x')
if side == 'both':
return sp.limit(self.symbolic_func, x, a)
elif side == 'left':
return sp.limit(self.symbolic_func, x, a, '-')
elif side == 'right':
return sp.limit(self.symbolic_func, x, a, '+')
class ContinuityAnalyzer:
def __init__(self, func, symbolic_func=None):
self.func = func
self.symbolic_func = symbolic_func
self.limit_analyzer = LimitAnalyzer(func, symbolic_func)
def check_continuity(self, a, tolerance=1e-10):
"""
Check if function is continuous at point a
Returns: (is_continuous, discontinuity_type, details)
"""
# Check if f(a) is defined
try:
fa = self.func(a)
fa_defined = True
except:
fa_defined = False
fa = None
# Compute one-sided limits
limit_results = self.limit_analyzer.numerical_limit(a)
# Extract final values (most accurate approximations)
if limit_results['left'] and not np.isnan(limit_results['left'][-1]):
left_limit = limit_results['left'][-1]
left_exists = True
else:
left_limit = None
left_exists = False
if limit_results['right'] and not np.isnan(limit_results['right'][-1]):
right_limit = limit_results['right'][-1]
right_exists = True
else:
right_limit = None
right_exists = False
# Determine continuity
if not fa_defined:
if left_exists and right_exists and abs(left_limit - right_limit) < tolerance:
return False, "removable", f"f({a}) undefined, but limits exist and agree"
else:
return False, "essential", f"f({a}) undefined"
if not (left_exists and right_exists):
return False, "essential", "One or both one-sided limits don't exist"
if abs(left_limit - right_limit) > tolerance:
return False, "jump", f"Left limit: {left_limit:.6f}, Right limit: {right_limit:.6f}"
limit_value = (left_limit + right_limit) / 2
if abs(fa - limit_value) > tolerance:
return False, "removable", f"f({a}) = {fa:.6f}, but limit = {limit_value:.6f}"
return True, "continuous", f"f({a}) = {fa:.6f} = limit"
def find_discontinuities(self, interval, num_test_points=100):
"""Find potential discontinuities in an interval"""
a, b = interval
test_points = np.linspace(a, b, num_test_points)
discontinuities = []
for point in test_points:
is_cont, disc_type, details = self.check_continuity(point)
if not is_cont:
discontinuities.append((point, disc_type, details))
return discontinuities
def plot_with_discontinuities(self, interval, num_points=1000):
"""Plot function highlighting discontinuities"""
a, b = interval
x = np.linspace(a, b, num_points)
y = []
for xi in x:
try:
y.append(self.func(xi))
except:
y.append(np.nan)
plt.figure(figsize=(12, 8))
plt.plot(x, y, 'b-', linewidth=2, label='f(x)')
# Find and mark discontinuities
discontinuities = self.find_discontinuities(interval, 50)
for point, disc_type, details in discontinuities:
color = {'removable': 'orange', 'jump': 'red', 'essential': 'purple'}.get(disc_type, 'black')
plt.axvline(x=point, color=color, linestyle='--', alpha=0.7)
plt.plot(point, 0, 'o', color=color, markersize=8,
label=f'{disc_type.title()} at x={point:.3f}')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.title('Function with Discontinuities Marked')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()
return discontinuities
def lhopital_demo():
"""Demonstrate L'Hôpital's rule for indeterminate forms"""
x = sp.Symbol('x')
examples = [
(sp.sin(x)/x, 0, "sin(x)/x as x→0"),
((sp.exp(x) - 1)/x, 0, "(e^x - 1)/x as x→0"),
(sp.log(x)/x, sp.oo, "ln(x)/x as x→∞"),
((1 - sp.cos(x))/x**2, 0, "(1 - cos(x))/x² as x→0"),
((x**2 - 1)/(x - 1), 1, "(x² - 1)/(x - 1) as x→1")
]
print("=== L'Hôpital's Rule Examples ===")
for expr, limit_point, description in examples:
print(f"\n{description}:")
# Direct limit
direct_limit = sp.limit(expr, x, limit_point)
print(f"Direct limit: {direct_limit}")
# Apply L'Hôpital's rule manually
numerator = sp.numer(expr)
denominator = sp.denom(expr)
# Check if it's an indeterminate form
num_limit = sp.limit(numerator, x, limit_point)
den_limit = sp.limit(denominator, x, limit_point)
if (num_limit == 0 and den_limit == 0) or (num_limit == sp.oo and den_limit == sp.oo):
# Apply L'Hôpital's rule
num_deriv = sp.diff(numerator, x)
den_deriv = sp.diff(denominator, x)
lhopital_expr = num_deriv / den_deriv
lhopital_limit = sp.limit(lhopital_expr, x, limit_point)
print(f"Numerator limit: {num_limit}")
print(f"Denominator limit: {den_limit}")
print(f"Indeterminate form: {num_limit}/{den_limit}")
print(f"After L'Hôpital's rule: {lhopital_limit}")
else:
print(f"Not an indeterminate form: {num_limit}/{den_limit} = {direct_limit}")
def epsilon_delta_demo(func, a, L, epsilon=0.1):
"""Demonstrate epsilon-delta definition of limits"""
print(f"=== ε-δ Demonstration ===")
print(f"Showing that lim(x→{a}) f(x) = {L}")
print(f"For ε = {epsilon}")
# Find δ such that |x - a| < δ ⟹ |f(x) - L| < ε
# This is a simplified approach for demonstration
delta_candidates = [0.1, 0.01, 0.001, 0.0001]
for delta in delta_candidates:
test_points = np.linspace(a - delta, a + delta, 21)
test_points = test_points[test_points != a] # Remove a itself
valid = True
for x in test_points:
try:
if abs(func(x) - L) >= epsilon:
valid = False
break
except:
valid = False
break
if valid:
print(f"δ = {delta} works: |x - {a}| < {delta} ⟹ |f(x) - {L}| < {epsilon}")
# Visualize
x_plot = np.linspace(a - 2*delta, a + 2*delta, 1000)
y_plot = [func(x) for x in x_plot]
plt.figure(figsize=(10, 6))
plt.plot(x_plot, y_plot, 'b-', linewidth=2, label='f(x)')
plt.axhline(y=L, color='r', linestyle='-', alpha=0.7, label=f'L = {L}')
plt.axhline(y=L+epsilon, color='r', linestyle='--', alpha=0.5, label=f'L ± ε')
plt.axhline(y=L-epsilon, color='r', linestyle='--', alpha=0.5)
plt.axvline(x=a, color='k', linestyle='-', alpha=0.7, label=f'x = {a}')
plt.axvline(x=a+delta, color='g', linestyle='--', alpha=0.7, label=f'x = a ± δ')
plt.axvline(x=a-delta, color='g', linestyle='--', alpha=0.7)
plt.fill_between([a-delta, a+delta], L-epsilon, L+epsilon, alpha=0.2, color='yellow')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.title(f'ε-δ Definition: ε = {epsilon}, δ = {delta}')
plt.legend()
plt.grid(True, alpha=0.3)
plt.show()
break
else:
print(f"δ = {delta} doesn't work")
# Example usage
if __name__ == "__main__":
print("=== Limit Analysis ===")
# Example 1: sin(x)/x at x = 0
def sinc(x):
return np.sin(x) / x if x != 0 else 1
sinc_symbolic = sp.sin(sp.Symbol('x')) / sp.Symbol('x')
analyzer1 = LimitAnalyzer(sinc, sinc_symbolic)
print("Function: sin(x)/x at x = 0")
symbolic_limit = analyzer1.symbolic_limit(0)
print(f"Symbolic limit: {symbolic_limit}")
numerical_result = analyzer1.numerical_limit(0)
print(f"Numerical left limit: {numerical_result['left'][-1]:.10f}")
print(f"Numerical right limit: {numerical_result['right'][-1]:.10f}")
print("\n=== Continuity Analysis ===")
# Example 2: Function with jump discontinuity
def step_func(x):
return 1 if x >= 0 else -1
continuity_analyzer = ContinuityAnalyzer(step_func)
is_cont, disc_type, details = continuity_analyzer.check_continuity(0)
print(f"Step function at x=0: {disc_type} - {details}")
# Example 3: Function with removable discontinuity
def removable_disc(x):
return (x**2 - 1) / (x - 1) if x != 1 else 0
continuity_analyzer2 = ContinuityAnalyzer(removable_disc)
is_cont, disc_type, details = continuity_analyzer2.check_continuity(1)
print(f"(x²-1)/(x-1) at x=1: {disc_type} - {details}")
print("\n=== L'Hôpital's Rule ===")
lhopital_demo()
print("\n=== ε-δ Demonstration ===")
epsilon_delta_demo(lambda x: 2*x + 1, 1, 3, 0.2)
9. Applications
9.1 Calculus
- Foundation for derivatives and integrals
- Optimization and related rates
- Fundamental Theorem of Calculus
9.2 Real Analysis
- Riemann integration theory
- Uniform convergence of function sequences
- Compactness and connectedness
9.3 Numerical Analysis
- Root finding algorithms
- Interpolation and approximation
- Error analysis
10. References
- Spivak, M. (2008). Calculus.
- Abbott, S. (2015). Understanding Analysis.
- Rudin, W. (1976). Principles of Mathematical Analysis.