[Math] Give an example of a continuous strictly increasing function g:R->R which is…

examples-counterexamplesreal-analysis

Give an example of a continuous strictly increasing function $g\colon \Bbb R\to \Bbb R$ which is differentiable at every $x$ not belonging to $\Bbb Z$ and not differentiable at any $x$ belonging to $\Bbb Z$.

Could I use a piecewise function to show this?

Best Answer

Here is a way to construct such a function.

  1. Draw $f(x)=x^3+x^2+x$.
  2. At every integer $n$, draw $g(x)$ between $n$ and $n+1$ as the secant from $f(n)$ to $f(n+1)$.
  3. $g(x)$ is now continuous, differentiable at every point between the integers, but not at the integers.

EDIT v2 I've edited to use a strictly increasing function.

Here is a graphical depiction of the concept:

enter image description here

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