I'm having a bit of trouble with the following problems. Any help would be greatly appreciated.
Are the following functions Lipschitz continuous near 0? If yes, find a Lipschitz constant for some interval containing 0.
a) f(x) = 1/(1-x^2)
For this one, I think it's Lipschitz near 0. I have
|f(x)-f(y)| = [(x+y)/(1-x^2)*(1-y^2)] * (x-y)
I'm not sure how to find the constant, though.
b) f(x) = x^2 sin(1/x)
I don't think this one is Lipschitz near 0, but I don't know how to show it.
If you show that both functions admit a continuous derivative in a closed interval containing 0, then you only have to apply the Mean Value Theorem to find such a constant.