Chapter 05

Continuity

Continuity at a point means the limit exists and equals the function value there. It is the basis for differentiability and for understanding activation and loss functions in deep learning.

Math diagram by chapter

Select a chapter to see its diagram below. View the flow of basic math at a glance.

Left: continuous — the curve passes through $(a, f(a))$ with no gap or jump. Right: discontinuous — a hole or jump at that point.

Continuous

lim = f(a)

Discontinuous

f(a) missing or lim ≠ f(a)

Continuity means $\lim_{x \to a} f(x) = f(a)$ . On the graph, the curve does not break at that point.

How to read

Left graph: $y = x^2$ is continuous at $x = 2$ (curve passes through (2, 4) with no break).
Right graph: if the function value is missing or differs from the limit at $x = a$ , it is discontinuous (hole or jump).

What is continuity?

Continuity at

x = a

means: as

x

approaches

a

f(x)

approaches $f(a)$ , and that limit equals

f(a)

. We write

\lim_{x \to a} f(x) = f(a)

. On the graph, the curve does not jump or break at that point.

In plain language: ①

f(a)

is defined, ②

\lim_{x \to a} f(x)

exists, and ③ that limit equals

f(a)

. If any of these fails, the function is discontinuous at

a

In ε-δ terms: For any small error

\varepsilon

, you can find a distance

\delta

so that if

x

is within

\delta

a

, then

f(x)

is within

\varepsilon

f(a)

. Same idea as in the limit chapter, but here the limit must equal $f(a)$ .

Differentiable implies continuous. For the derivative (instantaneous rate of change) to exist at a point, the function must have a value there and the limit must match it. So continuity is essential before studying derivatives.

In deep learning, activation functions (ReLU, sigmoid, etc.) and loss functions are usually continuous. When the input changes a little, the output should change smoothly so that gradient descent behaves stably.

In AI, the loss function measures how far the prediction is from the correct answer; it must be continuous so that small improvements lead to small decreases in loss. Activation functions are continuous (or piecewise continuous), so backprop and gradient computation are well-defined.

To check continuity at a point, verify: does $\lim_{x \to a} f(x)$ exist?, is $f(a)$ defined?, and are they equal?

Checklist: ①

f(a)

exists ②

\lim_{x \to a} f(x)

exists ③ limit

= f(a)

. If any fails, the function is discontinuous at that point.

Example problems and solutions are in the table below.

Problem	Solution
Ex 1. Is $f(x) = x^2$ continuous at $x = 2$ ?	Solution: $f(2) = 4$ , $\lim_{x \to 2} x^2 = 4$ ; they match, so continuous.
Ex 2. Is $g(x) = \frac{1}{x}$ continuous at $x = 0$ ?	Solution: $g(0)$ is not defined → discontinuous.
Ex 3. Is $h(x) = 2x + 1$ continuous at $x = -1$ ?	Solution: $h(-1) = -1$ , $\lim_{x \to -1} (2x+1) = -1$ ; they match, so continuous.