Figure

$f(0) = \sin 0 = 0$ , so

$f$ is defined at 0.

$\lim\limits_{x\rightarrow 0}f(x) = \lim\limits_{x\rightarrow 0}\sin x = 0$ , so the limit at 0 exists.

$\lim\limits_{x\rightarrow 0}\sin x = \sin 0 = 0$ .

Since all three conditions of continuity are satisfied, $f(x) = \sin x$ is continuous at 0.