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

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October 22nd 2009, 10:27 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. $
October 22nd 2009, 10:34 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 $x=0$ .
October 25th 2009, 08:05 PM
binkypoo

doesnt this show that f is continuous at x = 0? What about the rest of the domain?
October 25th 2009, 08:19 PM
redsoxfan325

It's continuous everywhere else because $x^2\sin(x)$ and $\frac{1}{x}$ are. The composition of continuous functions is continuous.
October 25th 2009, 08:33 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
October 25th 2009, 08:54 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.