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?

  • 2
    $\begingroup$ Every continuous function is a uniform limit of its piecewise linear interpolants. $\endgroup$ – user147263 Dec 3 '15 at 16:47
  • $\begingroup$ @NormalHuman I need it to be differentiable everywhere, not only continuous. $\endgroup$ – Farewell Dec 3 '15 at 16:48
  • $\begingroup$ Every differentiable function is continuous. $\endgroup$ – user147263 Dec 3 '15 at 16:48
  • $\begingroup$ That's a full answer, not a comment. $\endgroup$ – Justpassingby Dec 3 '15 at 16:50
  • $\begingroup$ @NormalHuman Of course, but you said that we can build continuous functions with the uniform limit of piecewise linear interpolants, i want to know can we build everywhere differentiable functions, so I do not see how your comments settle the question. $\endgroup$ – Farewell Dec 3 '15 at 16:52

Every continuous function $f$ on $[a,b]$ is a uniform limit of piecewise linear functions, which was proved here. The idea of proof is to partition $[a,b]$ into subintervals on each of which $f$ changes by at most $\epsilon/2$, which is possible by uniform continuity. Then the piecewise linear interpolation does the job.

In particular, every differentiable function can be written as a limit of piecewise linear functions.

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