Oh, well, the title actually describes what kind of question will this question be, but let us do some warm-up before stating the question as clearly as possible.
Suppose first that everything we do we do on some set of the form $[a,b]$.
Suppose secondly that we have some sequence $f_k$ ; $k \in \mathbb N$ of piecewise linear continuous functions defined on $[a,b]$:
Well, it seems to me, although I might be completely wrong, that when passing to the limit the only everywhere differentiable function which we can build from the sequence of piecewise linear continuous functions is the linear function $f(x)=cx+d$ ; $c,d \in \mathbb R$ or the constant function $f(x)=d$.
In other words, it seems to me that only the linear functions and constant functions will be everywhere differentiable functions that can arise from the limit of continuous piecewise linear functions.
So the question is, clearly:
Is there any other function that is differentiable everywhere on some set $[a,b]$ and such that that function is limit of continuous piecewise linear functions?