Continuous on a range

sin(x) is a continous function, cos(x) is a continous function, so you only need to verify that at $x = \frac{\pi}{4}$, $sin(\frac{\pi}{4}) = cos(\frac{\pi}{4})$. Why is this? Basically even though $\frac{\pi}{4}$ is not in the domain of sin(x), $lim_{x^{-} \to \frac{\pi}{4}} f(x) \to cos(\frac{\pi}{4})$ $lim_{x^{-} \to \frac{\pi}{4}} f(x) = lim_{x^{-} \to \frac{\pi}{4}} sin(x)$ Since sin(x) is continous $lim_{x^{-} \to \frac{\pi}{4}} sin(x) = sin(\frac{\pi}{4})$
In a gist, if you come at $x = \frac{\pi}{4}$ from the left, it should be the same as if you come at it from the right. The only point at which problem can arise is at $x = \frac{\pi}{4}$, which is why you need to check there.