Differentiability of composite functions

[raphile]

Hi, I have a small question about this. Using the chain rule, I know that a composition of differentiable functions is differentiable. But is it also true that if a composition of functions is differentiable, then all the functions in the composition must be differentiable?

For example, if f(g(x)) is differentiable, does that imply f(x) and g(x) are both differentiable?

Thanks!

 

[DonAntonio]

Hi, I have a small question about this. Using the chain rule, I know that a composition of differentiable functions is differentiable. But is it also true that if a composition of functions is differentiable, then all the functions in the composition must be differentiable?

For example, if f(g(x)) is differentiable, does that imply f(x) and g(x) are both differentiable?

Thanks!

\cos\sqrt{x} is differentiable from the right at x=0 , but \sqrt{x} isn't...

DonAntonio

 

[Office_Shredder]

Examples can be constructed where f(x) or g(x) can be arbitrarily bad (for example discontinuous) yet f(g(x)) is differentiable

A classic example of this is when f(x)=1 and g(x) is any function you pick. f(g(x))=1 so this composition is differentiable, but g(x) clearly doesn't have to be.

For the other way around consider f(x)="any function which is always negative" if x<1 (for example -x⁶), and x⁴ if x >=1. Now let g(x) = x²+3. Because g(x)>=1 always, f(g(x)) is always differentiable (because we are always using the x⁴ portion of f(x) thanks to how g was constructed), even though a large chunk of f(x) is not differentiable.

 

[raphile]

Thanks for the help and examples!