So I have another question about functions:

Question: If neither $f$ nor $g$ is differentiable at $a$, but both are continuous at $a$, then $f+g$ is not differentiable at $a$.

I know that we could have a function $f(a)=|a|$ and $g(a)=|a|$, where $a=0$, so this means $f$ and $g$ are both continuous but not differentiable at $a=0$.

But how do I show that $f+g$ is not differentiable now? How would I go about this?

  • 1
    $\begingroup$ You can't -- the (rather unsatisfying) counterexample is that $|x|$ and $-|x|$ are continuous every, nondifferentiable at the origin, but their sum is differentiable everywhere. $\endgroup$
    – Rellek
    Dec 16, 2017 at 19:22

1 Answer 1


Let $f=|x|$ and $g=-|x|$.

The statement is not true.

  • $\begingroup$ Thank you, would it still work if I used f(x) = |x| and g(x) = |x|? or does it have to be f(x) = |x| and g(x) = -|x|?? $\endgroup$ Dec 16, 2017 at 19:27
  • $\begingroup$ $f(x)+g(x)=2|x|$ it is still not differentiable at $x=0$ hence it can't be a counter example. $\endgroup$ Dec 16, 2017 at 19:28
  • $\begingroup$ Okay, thank you for the help. $\endgroup$ Dec 16, 2017 at 19:30

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