Checking the three-part definition of continuity at a point

First, evaluate the function at the point: by definition, $e(\pi/4) = 1$, so condition 1 is satisfied. Next, find the limit as $x \to \pi/4$. Since $e(x) = \tan(x)$ for all $x$ near $\pi/4$ except at the point itself, and $\tan(x)$ is continuous at $x=\pi/4$, we have $\lim_{x \to \pi/4} e(x) = \lim_{x \to \pi/4} \tan(x) = \tan(\pi/4) = 1$. So the limit exists and equals $1$, satisfying condition 2.

Now compare the limit to the function value: $\lim_{x \to \pi/4} e(x) = 1$ but $e(\pi/4) = 1$ as well... wait, checking definitions carefully, $e(\pi/4)$ is defined to be $1$, which happens to match the limit value $\tan(\pi/4)=1$. Since all three conditions hold ($e(\pi/4)$ defined, limit exists, and limit equals $e(\pi/4)$), the function $e(x)$ is continuous at $x = \pi/4$.