Question

Prove that if f is differentiable at a, then |f| is also differentiable at a, provided that f(a) is not 0.

Answer 1

if
\[f(a)\neq0\] then either \[|f(a)|=f(a)\] or
\[|f(a)|=-f(a)\]

Answer 2

if f is differentiable at a then -f is as well. done

Answer 3

u need to use precise defintion of a derivative

Answer 4

this is an incomplete proof

Answer 5

really? i suppose it is absurd to think that \[(-f(a))'=-f'(a)\]?

Answer 6

Actually, it's not absurd. It is quite right that if f is differentiable at a then -f is also differentiable at a. The only reason you need to have the f(a) not equal 0 is to prevent the graph from having a point where it would otherwise cross the x-axis.

Answer 7

just write out the definition of the derivative of -f'(a), pull out the - sing, assert that the thing after the - sign exists and be done

Answer 8

i suppose if you want to be really pedantic you could mention that if \[a\neq 0\] there is some \[\epsilon\] for which 0 is not contained in
\[(a-\epsilon.a+\epsilon) \]