Limits and Continuity

Calculus is built on the concept of change over zero. To make sense of “instantaneous” change, we need a way to talk about what a function does as it gets closer and closer to a point, even if it never actually reaches it. This is the Limit.

1. The Epsilon-Delta Definition

Intuition: A limit $\lim_{x \to c} f(x) = L$ means we can force $f(x)$ to be as close to $L$ as we want (within $\epsilon$), just by making $x$ sufficiently close to $c$ (within $\delta$).

For every $\epsilon > 0$, there exists a $\delta > 0$ such that $0 < |x - c| < \delta \implies |f(x) - L| < \epsilon$.

Let’s vizualize this. If $f(x) = 2x$, and we want to be within $\epsilon = 0.1$ of the output $L=4$ at $x=2$, how close must $x$ be?

import numpy as np
import matplotlib.pyplot as plt

def f(x): return 2 * x

c, L = 2, 4
epsilon = 0.1
delta = epsilon / 2 # For f(x)=2x, delta is exactly epsilon/2

x = np.linspace(c - 2*delta, c + 2*delta, 500)
y = f(x)

plt.plot(x, y, label='f(x)=2x')
plt.axhline(L + epsilon, color='r', linestyle='--', label='L + ε')
plt.axhline(L - epsilon, color='r', linestyle='--', label='L - ε')
plt.axvline(c + delta, color='g', linestyle='--', label='c + δ')
plt.axvline(c - delta, color='g', linestyle='--', label='c - δ')

plt.fill_between([c-delta, c+delta], L-epsilon, L+epsilon, color='yellow', alpha=0.3, label='Safe Zone')
plt.legend()
plt.title("The Epsilon-Delta Game")
plt.show()

2. When Limits Fail

A limit only exists if it matches from both directions. If the “safe zone” can’t be established because the function jumps or oscillates, the limit does not exist.

3. Continuity: The Glue of Calculus

A function is continuous at $c$ if the limit exists and exactly matches the function’s value: $\lim_{x \to c} f(x) = f(c)$.

Let’s test for a “Broken” function: $f(x) = \begin{cases} x^2 & x < 1 \\ 2 & x = 1 \\ x & x > 1 \end{cases}$

import numpy as np

def f(x):
  if x < 1: return x**2
  if x == 1: return 2
  return x

# Check limit from left and right
left_lim = f(0.999999)
right_lim = f(1.000001)
value = f(1.0)

print(f"Limit from left: {left_lim}")
print(f"Limit from right: {right_lim}")
print(f"Actual value at 1: {value}")

is_continuous = np.isclose(left_lim, right_lim) and np.isclose(left_lim, value)
print(f"Is continuous? {is_continuous}")

4. The Intermediate Value Theorem (IVT)

If $f$ is continuous on $[a, b]$, it must hit every value between $f(a)$ and $f(b)$. This is why we can use the Bisection Method to find roots.

5. Summary Check