Thread: continuous function proof

Suppose that f(x) is any function that is continuous at x=0.
Prove that the function g(x) = xf(x) is differentiable at x=0.

Now my first instinct was to say that this cannot be done.
If a function is continuous at a certain value of x does not mean its differentiable at that point as well.

Can someone shed some light here.

Maybe the question intends for the function to be differentiable at (x) as well.

Go back to the definition: g(x) is differentiable at 0 if $\displaystyle \lim_{h\to0}\frac{g(h)-g(0)}h$ exists. So, does that limit exist, or not?

Maybe the question intends for the function to be differentiable at (x) as well.

Probably, because continuity doesn't imply differentiability. However, if you can take a derivative at a point, then it does mean the function is continuous at that point.

it could, depending on the function

|x| for example