Intermediate value property problem and continuous function

Let $c$ be a point with $a<c<b$, and suppose given a small $\epsilon > 0$ (small enough that $c+\epsilon<d$ and otherwise arbitrarily small). Use the intermediate value property (and strict increasingness of $f$) to pick $d$ with $c<d<b$ 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.