# analysis of derivatives

• Apr 23rd 2009, 12:18 PM
noles2188
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.
• Apr 23rd 2009, 04:33 PM
putnam120
For 2:
Consider the sequence $\{x_n\}$ such that $x_n\to x$.
If $x\neq{0}$ then $f(x_n)\not\to f(x)$. This is because if x is rational take each $x_n$ 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 $\displaystyle\frac{f(x_n)}{x_n}$ equals 0 is $x_n$ is irrational and 0 if $x_n$ is rational. So from this we see that $f'(0)=0$.