Intermediate value property problem and continuous function How to prove that a strictly increasing function $f:[a, b]\rightarrow \mathbb{R}$ which has the intermediate value property is continuous on $[a, b]$.
Let $c$ be a point with $a&lt;c&lt;b$, and suppose given a small $\epsilon &gt; 0$ (small enough that $c+\epsilon&lt;d$ and otherwise arbitrarily small). Use the intermediate value property (and strict increasingness of $f$) to pick $d$ with $c&lt;d&lt;b$ for which $f(d)=f(c)+\epsilon.$  Since $f$ is strictly increasing, for all $x$ in $[c,d]$ we have $$f(c) \le f(x) \le f(d).$$ And $f(d)-f(c)=\epsilon.$ [Note: In the $\epsilon, \delta$ continuity definition, we have chosen $\epsilon$ first, and then found the point $d$, and are defining $\delta=d-c$.]  This given "half" of continuity at $c$, and the left half is similar to show.

At either endpoint $a$ or $b$ we only need the above argument one way.
You cannot prove that because it is not true, there are functions that are not continuous that have this property. Just think of a function that is strictly increasing but with discontinuities and that is defined in the hole interval that you will notice that it can still obey the property.