continuous function proof

**chrisc** · Dec 5th 2007, 11:50 AM

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.

**Opalg** · Dec 5th 2007, 12:09 PM

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

**colby2152** · Dec 5th 2007, 12:24 PM

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.

**chrisc** · Dec 5th 2007, 12:26 PM

it could, depending on the function

|x| for example

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