$\alpha$ monotonically increasing on [a, b] $\implies$ finite at $a$ and $b$, why? The following statement is part of the definition 6.2 in Baby Rudin: > Let $\alpha$ be a monotonically increasing function on $[a,b]$ ( **since $\alpha (a)$ and $\alpha (b)$ are finite** , it follows that $\alpha$ is bounded on $[a,b]$). Why is the part in bold necessarily true? Isn't $\tan (x)$ on $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ monotonically increasing but infinite at the endpoints?

If $f$ is a real-valued function on $[a,b]$, then $f(a)$ and $f(b)$ are real numbers and hence finite.

The function $\tan{x}$ isn't defined at $\pm\pi/2$, so its domain isn't on $[-\pi/2, +\pi/2]$.