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

• Oct 22nd 2009, 11:27 PM
binkypoo
how can I prove if this function is continuous or not!
$
f(x) = \left\{
\begin{array}{lr}
x^2 sin\frac{1}{x} & : x \neq 0\\
0 & : x = 0
\end{array}
\right.
$
• Oct 22nd 2009, 11:34 PM
redsoxfan325
Quote:

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

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

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

It's also differentiable at $x=0$.
• Oct 25th 2009, 09:05 PM
binkypoo
doesnt this show that f is continuous at x = 0? What about the rest of the domain?
• Oct 25th 2009, 09:19 PM
redsoxfan325
It's continuous everywhere else because $x^2\sin(x)$ and $\frac{1}{x}$ are. The composition of continuous functions is continuous.
• Oct 25th 2009, 09:33 PM
binkypoo
I guess my problem is I dont know how to show that $x^2sin(x)$ and 1/x are continuous EVERYWHERE. Could you explain that please? thanks
• Oct 25th 2009, 09:54 PM
redsoxfan325
Quote:

Originally Posted by binkypoo
I guess my problem is I dont know how to show that $x^2sin(x)$ 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 $\epsilon$- $\delta$ proofs for each one.

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

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

$\frac{1}{x}$ 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.