Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 30

Locate the discontinuities of the function $ y = \tan \sqrt{x}$ and illustrate its graph.

We can rewrite the function $y = \tan \sqrt{x}$ as $\displaystyle y = \frac{\sin \sqrt{x}}{\cos \sqrt{x}}$ and locate its discontinuity where its denominator is 0.


$
\begin{equation}
\begin{aligned}

\cos \sqrt{x} &= 0\\
\sqrt{x} &= \cos^{-1} [0]\\
\sqrt{x} &= \frac{\pi}{2} + \pi n \qquad ;\text{where } n \text{ is a positive integer because the root function is defined only for positive values of } x
\end{aligned}
\end{equation}
$

Therefore,
$\quad$ The function $y = \tan \sqrt{x}$ is discontinuous at $\displaystyle x = \left(\frac{\pi}{2} + \pi n \right)^2$; where $n$ is a positive integer.