# how can I prove if this function is continuous or not!

Printable View

• Oct 22nd 2009, 10:27 PM
binkypoo
how can I prove if this function is continuous or not!
$\displaystyle f(x) = \left\{ \begin{array}{lr} x^2 sin\frac{1}{x} & : x \neq 0\\ 0 & : x = 0 \end{array} \right.$
• Oct 22nd 2009, 10:34 PM
redsoxfan325
Quote:

Originally Posted by binkypoo
$\displaystyle f(x) = \left\{ \begin{array}{lr} x^2 sin\frac{1}{x} & : x \neq 0\\ 0 & : x = 0 \end{array} \right.$

Take the limit: $\displaystyle -\lim_{x\to0}x^2\leq\lim_{x\to0}x^2\sin(1/x)\leq\lim_{x\to0}x^2$

So $\displaystyle \lim_{x\to0}x^2\sin(1/x)=0=f(0)$, and it's continuous at $\displaystyle x=0$.

It's also differentiable at $\displaystyle x=0$.
• Oct 25th 2009, 08:05 PM
binkypoo
doesnt this show that f is continuous at x = 0? What about the rest of the domain?
• Oct 25th 2009, 08:19 PM
redsoxfan325
It's continuous everywhere else because $\displaystyle x^2\sin(x)$ and $\displaystyle \frac{1}{x}$ are. The composition of continuous functions is continuous.
• Oct 25th 2009, 08:33 PM
binkypoo
I guess my problem is I dont know how to show that $\displaystyle x^2sin(x)$ and 1/x are continuous EVERYWHERE. Could you explain that please? thanks
• Oct 25th 2009, 08:54 PM
redsoxfan325
Quote:

Originally Posted by binkypoo
I guess my problem is I dont know how to show that $\displaystyle x^2sin(x)$ and 1/x are continuous EVERYWHERE. Could you explain that please? thanks

I don't think your professor expects you to do all that, but if he does, you need to crank out the $\displaystyle \epsilon$-$\displaystyle \delta$ proofs for each one.

For $\displaystyle x^2$ let $\displaystyle \delta=\min\left\{1,\frac{1}{1+2|x_0|}\right\}$, because $\displaystyle |x-x_0|<1\implies |x+x_0|<1+2|x_0|\implies |x-x_0||x+x_0|<\delta(1+2|x_0|)$

For $\displaystyle \sin x$, you can probably Google a proof. The idea is to let $\displaystyle \delta=\epsilon$ and then use a difference-to-product identity to convert $\displaystyle |\sin x-\sin x_0|$ into something more manageable.

$\displaystyle \frac{1}{x}$ is kind of a pain in the ass, and I've screwed up continuity proofs of this function before. There are probably other threads on MHF with proofs of this.