Prove that if $|f|$ is differentiable at $a$ and $f$ is continuous at $a$, then $f$ is differentiable at $a$

Question

Prove that if $|f|$ is differentiable at $a$ and $f$ is continuous at $a$, then $f$ is differentiable at $a$.

We are given $\displaystyle \lim_{x \to a}\dfrac{|f(x)|-|f(a)|}{x-a} = c$ and we know $f$ is continuous at $a$. We must prove that $\displaystyle \lim_{x \to a} \dfrac{f(x)-f(a)}{x-a} = b$. Using the fact that $f$ is continuous at $a$ gives us $$\forall \epsilon, \exists \delta \quad 0<|x-a|<\delta \quad \implies \quad |f(x)-f(a)| < \epsilon.$$ I am not sure how to use these facts to prove the statement.

If $f(a)>0$ or $f(a)<0$, there is almost nothing to prove. So you can safely assume $f(a)=0$. — Alex R., Mar 25 '16 at 17:50

score 4 · Accepted Answer · 2016-04-20T14:19:17.307

4

Because $f$ is continues, if $f(a)>0$ or $f(a)<0$ then $f=|f|$ or $-|f|$ in the neighborhood of $a$, respectively, and so $f$ is differentiable at $a$.

If $f(a)=0$ then $|f(a)|'=0$, because $|f|$ is differentiable at $a$ and $a$ is the minimum point of $|f|$. Thus $f'(a)=|f(a)|'=0$ and so $f$ is differentiable at $a$.

edited Apr 20 '16 at 14:19

answered Mar 25 '16 at 17:59

How if $f(a) = 0$ then $|f(a)|'=0$? – user19405892 Mar 25 '16 at 18:14
@user19405892 I wrote the reason. – Mar 25 '16 at 18:16

Prove that if $|f|$ is differentiable at $a$ and $f$ is continuous at $a$, then $f$ is differentiable at $a$

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