Continuity: When Nearby Inputs Stay Nearby

Continuity formalizes a familiar expectation: a small change in the input should not produce an uncontrolled jump in the output. The idea sounds visual, but its real power comes from translating that picture into a statement about limits.

The three conditions

A function $f$ is continuous at $x=a$ when all three statements hold:

1. $f(a)$ is defined.
2. $\lim_{x\to a} f(x)$ exists.
3. $\lim_{x\to a} f(x)=f(a)$.

Why continuity matters

Continuity rules out sudden local surprises. It is the condition behind many existence results: a continuous path between values cannot skip every value in between.

Repair and use continuity

For $f(x)=(x^2-4)/(x-2)$ when $x\ne2$, cancellation gives $f(x)=x+2$, so the limit at $2$ is $4$. Defining $f(2)=4$ repairs the hole; any other value leaves a discontinuity.

Check your understanding

Why is $1/x$ not “discontinuous at zero” in the pointwise definition?

Show the reasoning

Zero is not in its domain, so continuity at zero is not defined. The function is continuous at every point in its domain.