10.98 min read

Continuity

A function $f$ is continuous at $x_0$ if three things hold: (1) $f$ is defined at $x_0$ ; (2) the limit $\lim_{x \to x_0} f(x)$ exists; and (3) the limit equals the function value: $\lim_{x \to x_0} f(x) = f(x_0)$ . Intuitively, continuity means you can draw the graph without lifting your pen.

Continuity fails in three ways: a removable discontinuity (limit exists but $f(x_0)$ is wrong), a jump discontinuity (left and right limits exist but differ), or an infinite/oscillating discontinuity (limit does not exist). A function is continuous on an interval if it is continuous at every point in that interval.

The class of continuous functions is called $C^0$ (or $\mathcal{C}^0$ ). It is the most basic regularity class — functions "nice enough" for optimization at least need to be continuous.

Formal View

Definition 10.8 — Continuity

A function

f

is continuous at $x_0$ if

\lim_{x \to x_0} f(x) = f(x_0)

. It is continuous on a set $D$ if it is continuous at every point of

D

. The class of continuous functions on

D

is denoted

C^0(D)

(or simply

C^0

Polynomials, $\sin$ , $\cos$ , $e^x$ , $\ln(x)$ (for $x>0$ ) are all continuous on their natural domains.

Interactive Visualization

Secant → Tangent Line

Why This Matters

Continuity is the minimum regularity needed for optimization — discontinuous functions can have minimizers that are impossible to find by descent methods.

Extreme Value Theorem: a continuous function on a closed bounded interval always attains its minimum and maximum.
Neural network activations must be continuous to allow gradient-based training (mostly — ReLU has a single discontinuity in derivative, not the function itself).
Physical models: conservation laws ensure physical quantities are continuous in space and time.

Learning Resources

Continuity and discontinuities

Khan Academy

Intuitive explanation of continuity and the three types of discontinuities.

11 min

Continuity formally

MIT OpenCourseWare

Rigorous treatment of continuity from MIT 18.01.

45 min

Quiz

Question 1

Which condition is NOT required for $f$ to be continuous at $x_0$ ?

Question 2

Every polynomial is a continuous function.

Common Mistakes

Thinking differentiability implies continuity — it does! But continuity does NOT imply differentiability.
Confusing a removable discontinuity (limit exists, value is wrong) with a jump discontinuity (one-sided limits differ).