Understanding a Fourier analysis problem

Question

I have the following problem from my Fourier analysis book:

where $PC(a,b)$ is the set of piecewise-continuous functions. I don't quite understand my task in this problem, what am I supposed to do? =)

thank you for any help!

Answer 1

You're supposed to cook up a strategy to answer this challenge: If I give you a piecewise-continuous function $f$ on $[a,b]$, you have to hand me back a sequence $f_k$ of continuous functions that converges to $f$ in the given norm. The picture is a hint as to how to construct the $f_k$.