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Analysis - Differentiation

Let f,g be functions such that the product h(x)=f(x)g(x) is differentiable at a. Does it follow that f and g are differentiable at a?
Is it possible that only one of f and g are differentiable at a?
Reply 1
Avatar for Wrangler
Wrangler
2
Zagani
Let f,g be functions such that the product h(x)=f(x)g(x) is differentiable at a. Does it follow that f and g are differentiable at a?

Try thinking of a few counterexamples (there are a lot). Here's a small [huge] hint - is √x differentiable at 0?
Reply 2
Avatar for k10k
k10k
6
Hmm is f(x)=?x a function?
Reply 3
Avatar for Jonny W
Jonny W
8
Select to read:

(1)
Let f be the zero function and g be something nasty.

(2)
Let

f(x)
=
0, if x is rational
1, if x is irrational

g(x)
=
1, if x is rational
0, if x is irrational

Then h(x) = 0 for all x, so h is differentiable everywhere.

But f and g are not differentiable anywhere.
Reply 4
Avatar for john!!
john!!
1
clever
Reply 5
Avatar for Wrangler
Wrangler
2
It might be worth putting on spoiler on those JonnyW, just incase the OP doesn't want the game totally given away! :smile:

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