# Math Help - Prove f continuous at a point

1. ## Prove f continuous at a point

f:[0,1]--->R given by
f(x)=0 if x=0
$f(x)=xcos(1/x)$, otherwise
How do i prove that f is continuous at the point a=0?

I tried to use the $\lim_{x\to0}f(x)=f(0)$ but I can't go any further from $\lim_{x\to0}cos(1/x)$. maybe I have to use epsilon/delta definition to solve this?

thanks

2. Why " $\lim_{x\to 0} cos(1/x)$" (which does not exist anyway)? The function you are concerned with is $x cos(1/x)$.

Since $-1\le cos(1/x)\le 1$ no matter what x is, $-x\le x cos(1/x)\le x$ for all x.

3. Is this correct?

$\lim_{x\to0}|x{cos{(1/x)}}|\le\lim_{x\to0}|x|=f(0)=0$