Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 36

Show that $f(x) = \left\{
\begin{array}{c}
\sin x & \text{ if } & x < \frac{\pi}{4} \\
\cos x & \text{ if } & x \neq \frac{\pi}{4}
\end{array} \right. $ is continuous on $(-\infty, \infty)$


Based from the definition,
A function $f$ is continuous at a number if and only if its left and right hand limits are equal. So,

$
\begin{equation}
\begin{aligned}
\lim\limits_{x \to \frac{\pi}{4}^-} x & = \lim\limits_{x \to \frac{\pi}{4}^+} \cos x\\
\sin \frac{\pi}{4} & = \cos \frac{\pi}{4}\\
\frac{\sqrt{2}}{2} & = \frac{\sqrt{2}}{2}
\end{aligned}
\end{equation}
$


Therefore,
$f$ is continuous on interval $(-\infty,-\infty)$