Hi all
I ask for your opinion on my solution to the following problem i have been trying to solve for too long (3~ hours):
"let w(x) be continuous for all x.
we define u(x) = 2+x^2+xw(x)
prove that u has a derivative at x=0."
So i did it like this:
u'(x) = 2x+w(x)+xw'(x)
u'(0) = w(0)
Since w is continuous for all x, w(0) exists, therefore u'(0) exists.
I don't know why but it doesn't "feel" complete as a proof...it was "too easy" and something seems to be missing im not sure what.
is it necessary to add that "the limit of u' as x approaches 0 = w(0) = u'(0)"?
Thank you for your help.
Nevermind, Plato said it better than I did.
The Weierstrass function - Wikipedia, the free encyclopedia is a famous example of a function that is continuous everywhere but differentiable nowhere
u'(x) = 2x+w(x)+xw'(x)
When you put x=0 into this expression for u'(x) you assumed that xw'(x) was zero but if w'(0) is undefined then 0*w'(0) is undefined.
If the question is insisting that you prove it then write a little stating that it is impossible to prove. That knowing a function is continuous does not mean it is differentiable. Mention the Weierstrass function, your lecturer has probably heard of it.