# Continuous function

#### Yankel

##### Active member
Hello

I need some help with this question please:

For which values of a the next function is continuous at x=0 ?

$$\left\{\begin{matrix} x^{a}\cdot sin\frac{1}{x} & x\neq 0\\ 0 & x=0 \end{matrix}\right.$$

I know that for it to be continuous at x=0, I need f(0)=lim x-->0

So I tried calculating the limit, and got to:

$$\lim_{x\to0}x^{a-1}\cdot sin\frac{1}{x}\cdot x$$

not sure I am correct, but anyhow do not know how to proceed. What I tried to do was to bring the limit to a known form

#### Ackbach

##### Indicium Physicus
Staff member
I would think about the squeeze theorem, and the fact that $-1\le \sin(x)\le 1\;\forall\,x$.