Answer
$f$ is continuous at $a$ if and only if $\lim\limits_{h \to 0}f(a+h) = f(a)$
Work Step by Step
Let's assume that $f$ is continuous at $a$
Then $\lim\limits_{x \to a}f(x) = f(a)$
Choose any $\epsilon \gt 0$
There is a positive number $\delta$ such that if $0 \lt \vert x-a \vert \lt \delta$ then $\vert f(x) -f(a) \vert \lt \epsilon$
If $0 \lt \vert h-0 \vert \lt \delta$ then $0 \lt \vert (a+h)-a \vert \lt \delta$ and $\vert f(a+h) -f(a) \vert \lt \epsilon$
Then $\lim\limits_{h \to 0}f(a+h) = f(a)$
Now let's assume that $\lim\limits_{h \to 0}f(a+h) = f(a)$
Choose any $\epsilon \gt 0$
There is a positive number $\delta$ such that if $0 \lt \vert h-0 \vert \lt \delta$ then $\vert f(a+h) -f(a) \vert \lt \epsilon$
If $0 \lt \vert x-a \vert \lt \delta$ then $\vert f(a+x-a) -f(a) \vert = \vert f(x) -f(a) \vert \lt \epsilon$
Then $\lim\limits_{x \to a}f(x) = f(a)$ and $f$ is continuous at $a$
Therefore:
$f$ is continuous at $a$ if and only if $\lim\limits_{h \to 0}f(a+h) = f(a)$
