Limits and Continuity

Analysis is the branch of mathematics that provides the rigorous foundation for calculus. While introductory calculus often relies on intuitive “infinitesimal” arguments, real analysis formalizes these using the language of limits.

The Formal Definition: $ϵ-\delta$

We say that the limit of $f(x)$ as $x$ approaches $c$ is $L$, written as ${\lim }_{x\to c}f(x)=L$, if for every $ϵ>0$, there exists a $\delta >0$ such that: $0<|x-c|<\delta ⟹|f(x)-L|<ϵ$

This definition captures the idea that we can get $f(x)$ as close as we want to $L$ ($ϵ$) by making $x$ close enough to $c$ ($\delta$), without $x$ actually having to equal $c$.

One-Sided Limits

A limit exists if and only if both the left-hand limit (${\lim }_{x\to c^{-}}$) and the right-hand limit (${\lim }_{x\to c^{+}}$) exist and are equal.

• If $f(x)=|x|/x$, then ${\lim }_{x\to 0^{+}}=1$ and ${\lim }_{x\to 0^{-}}=-1$. The two-sided limit does not exist.

Continuity

A function $f:D\to \mathbb{R}$ is continuous at a point $c\in D$ if:

1. $f(c)$ is defined.
2. ${\lim }_{x\to c}f(x)$ exists.
3. ${\lim }_{x\to c}f(x)=f(c)$.

In the $ϵ-\delta$ language, $f$ is continuous at $c$ if for every $ϵ>0$, there exists a $\delta >0$ such that $|x-c|<\delta ⟹|f(x)-f(c)|<ϵ$. Note that we no longer require $0<|x-c|$, because the value at $c$ is now part of the condition.

Key Theorems of Continuous Functions

Continuous functions on closed intervals $[a,b]$ possess powerful properties:

1. The Intermediate Value Theorem (IVT)

If $f$ is continuous on $[a,b]$ and $y$ is any value between $f(a)$ and $f(b)$, then there exists at least one $c\in (a,b)$ such that $f(c)=y$.

• Application: This provides a rigorous proof that polynomials of odd degree always have at least one real root.

2. The Extreme Value Theorem (EVT)

If $f$ is continuous on a closed and bounded interval $[a,b]$, then $f$ attains a maximum and a minimum value on that interval.

• Significance: This is the theoretical basis for optimization in calculus.

Uniform Continuity

A function $f$ is uniformly continuous on a set $A$ if for every $ϵ>0$, there exists a $\delta >0$ such that for all $x,y\in A$: $|x-y|<\delta ⟹|f(x)-f(y)|<ϵ$ The key difference from ordinary continuity is that $\delta$ depends only on $ϵ$, not on the point $x$.

• $f(x)=x^2$ is continuous on $\mathbb{R}$, but not uniformly continuous on $\mathbb{R}$ (because it gets steeper as $x\to \infty$).
• Heine-Cantor Theorem: If $f$ is continuous on a compact set (e.g., $[a,b]$), then $f$ is uniformly continuous on that set.

Sequences and Limits at Infinity

In analysis, we often consider the limit of a sequence $(a_n)$ as $n\to \infty$. A sequence converges to $L$ if for every $ϵ>0$, there exists an $N\in \mathbb{N}$ such that: $n>N⟹|a_n-L|<ϵ$ Functions have limits at infinity if $f(x)$ approaches a value as $x\to \infty$ or $x\to -\infty$, represented by horizontal asymptotes.

Python: Visualizing $ϵ-\delta$

We can write a script to find a suitable $\delta$ for a given $f$, $c$, and $ϵ$.

import math

def f(x):
    return 3 * x + 2

def find_delta(c, epsilon, test_range=0.1, steps=1000):
    L = f(c)
    best_delta = 0
    # Search for the largest delta that satisfies the condition
    for d in [i * (test_range/steps) for i in range(1, steps)]:
        # Check points in (c-d, c+d)
        is_valid = True
        for test_x in [c - d, c + d]:
            if abs(f(test_x) - L) >= epsilon:
                is_valid = False
                break
        if is_valid:
            best_delta = d
        else:
            break
    return best_delta

c = 2
epsilon = 0.01
delta = find_delta(c, epsilon)
print(f"For epsilon={epsilon}, a valid delta is: {delta}")
# For f(x) = 3x + 2, we expect delta to be epsilon/3 ≈ 0.0033

Summary

The study of limits is the study of approximation. By mastering the $ϵ-\delta$ definition, we move from the “hand-waving” of early calculus to the absolute precision of modern analysis. This precision allows us to handle complex phenomena like fractals, non-differentiable functions, and infinite series without falling into logical paradoxes.

Exercise