Real analysis of intermediate value property

Question

How to prove that a strictly increasing function f:[a,b]→R which has the intermediate value property is continuous on [a,b]

Answer 1

Pick your favorite point $c \in (a,b)$.

Since $f$ is increasing, both one-sided limits $$\lim_{x \to c^-} f(x) \quad \text{and} \lim_{x \to c^+} f(x)$$ exist.

It is evident that $\displaystyle \lim_{x \to c^-} f(x) \le \lim_{x \to c^+} f(x)$. You should show that $\displaystyle \lim_{x \to c^-} f(x) < \lim_{x \to c^+} f(x)$ will violate the intermediate value property. This implies $f$ is continuous at $c$.

The same argument works at the endpoints, just replace the relevant one-sided limits with $f(a)$ or $f(b)$.