Answer
$f$ is continuous on $(-\infty,\infty)$.
Work Step by Step
\[f(x_=\left\{\begin{array}{ll}\sin x\;\;\;\;,x< \frac{\pi}{4}\\
\cos x\;\;\;\;,x\geq\frac{\pi}{4}\end{array}\right.\]
Clearly $f$ is continuous if $x\neq \frac{\pi}{4}$, so we will check the continuity at $x=\frac{\pi}{4}$
\[\lim_{x\rightarrow \frac{\pi}{4}^{-}}f(x)=\lim_{x\rightarrow \frac{\pi}{4}}\sin x=\sin\frac{\pi}{4}=\frac{1}{\sqrt{2}}\]
\[\lim_{x\rightarrow \frac{\pi}{4}^{+}}f(x)=\lim_{x\rightarrow\frac{\pi}{4}}\cos x=\cos \frac{\pi}{4}=\frac{!}{\sqrt{2}}\]
\[\Rightarrow \lim_{x\rightarrow \frac{\pi}{4}^{-}}f(x)=\lim_{x\rightarrow \frac{\pi}{4}^{+}}f(x)=f(\frac{\pi}{4})\]
$\Rightarrow f$ is continuous at $x=\frac{\pi}{4}$
Therefore $f$ is continuous on $(-\infty,\infty)$
Hence Proved.
