Mathematics

Continuity at a point hinges on whether the left and right limits match the function value there. Here, you need to check if $\lim_{x \to 0^-} \sin x$, $\lim_{x \to 0^+} x$, and $f(0)$ all agree. Since $\sin x$ approaches $0$ as $x \to 0^-$ and $x$ approaches $0$ as $x \to 0^+$, while $f(0) = 0$ by definition, all three match—so the function is continuous at $x=0$.

Continuity at a point hinges on whether the left and right limits match the function value there. Here, you need to check if $\lim_{x \to 0^-} \sin x$, $\lim_{x \to 0^+} x$, and $f(0)$ all agree. Since $\sin x$ approaches $0$ as $x \to 0^-$ and $x$ approaches $0$ as $x \to 0^+$, while $f(0) = 0$ by definition, all three match—so the function is continuous at $x=0$.