continuous

A continuous function is one where there are no "breaks" or "jumps" in the graph.

Formally, a function $f$ is continuous at a point a if:

$$\begin{array}{rl}\underset{x\to a}{\lim }f(x)& =f(a)\end{array}$$

…the limit of $f(x)$ as $x$ approaches a equals $f(a)$.

Continuity $\ne$ Differentiability

If a function is differentiable at a point, then it must be continuous at that point.
However, a function can be continuous at a point without being differentiable there.
E.g. $f(x)=|x|$ is continuous everywhere but not differentiable at $x=0$; $f(x)=x^{1/3}$ has a vertical tangent at $x=0$, i.e. not differentiable there.

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