Show that there exists a step function Suppose $f:[a,b] \rightarrow X$ is a continuous map. By an argument based on uniform continuity, show that for any $\epsilon>0$ there exists a step-function $u: [a,b] \rightarrow X$ such that for all $x \in [a,b]$, $||f(x)-u(x)||<\epsilon$.
Okay, so I have looked back to the uniform continuity-definition, but I still can't see how I can connect that definition with the exercise...
Any help would be highly appreciated...

We know that $x-\lfloor{x}\rfloor\leq1$. Using this fact, 
$$u(x)=\epsilon\cdot\lfloor \frac{f(x)}{\epsilon}\rfloor$$
