Continuity With Piecewise Functions and Limits

Question

Can someone explain how I would do this problem?

Answer 1

Hint: $$\lim_{x\rightarrow c^-}f(x)=\lim_{x\rightarrow c^+}f(x)$$ Where will $\sin c=\cos c ?$ Or better.
$\tan c=1$

Answer 2

$\sin x$ and $\cos x$ are both continuous functions, so the only place your function $f(x)$ might not be continuous is at the point $x=c$, when it switches from being $\sin x$ to $\cos x$.

To ensure $f(x)$ is continuous at $c$, we need $$\lim_{x \to c} f(x)=f(c)$$ i.e., $$\lim_{x \to c^-} f(x)=\lim_{x \to c^+}f(x) = f(c).$$ Now you should be able to fill in what those two limits are and what $f(c)$ is, based on the definition of $f(x)$. What does that tell you about $c$?