Answer
Take $h=x-c$ and rewrite the definition of continuity into:
$f$ is continuous at $c$ if and only if $$\lim_{h\to0}f(c+h)=f(c)$$
Work Step by Step
According to the definition of continuity, $f$ is continuous at $c$ if and only if $$\lim_{x\to c}f(x)=f(c)$$
Now let's take $h=x-c$, meaning that $x=c+h$
So as $x$ approaches $c$, $h=x-c$ will approach $0$.
Therefore, the definition of continuity can be written as follows:
$f$ is continuous at $c$ if and only if $$\lim_{h\to0}f(c+h)=f(c)$$
