Thread: analysis of derivatives

1) Let f(x)=x^2 sin(1/x^2) for x not equal to 0 and f(0)=0.

a)Show a is differentiable on R
b)Show that f ' is not bounded on the interval [-1,1].

2) Let f(x)=x^2 if x is rational and f(x)=0 if x is irrational.

a)Prove that f is continuous at exactly one point, namely at x=0
b)Prove that f is differentiable at exactly one point, namely at x=0.

For 2:
Consider the sequence $\displaystyle \{x_n\}$ such that $\displaystyle x_n\to x $.
If $\displaystyle x\neq{0}$ then $\displaystyle f(x_n)\not\to f(x)$. This is because if x is rational take each $\displaystyle x_n$ to be irrational, and vice-versa if x is irrational.

For the second part, since f is only continuous at 0, this is the only point that it can possibly be differentiable. Now $\displaystyle \displaystyle\frac{f(x_n)}{x_n}$ equals 0 is $\displaystyle x_n$ is irrational and 0 if $\displaystyle x_n$ is rational. So from this we see that $\displaystyle f'(0)=0$.