example of a function that is continuous but not differentiable

**dgmath** · April 22nd 2010, 05:22 PM

I need an example of a function that is continuous at some x0 but not differentiable at x0.

is f(x) = |x| a right answer?

**Drexel28** · April 22nd 2010, 05:35 PM

Originally Posted by dgmath

I need an example of a function that is continuous at some x0 but not differentiable at x0.

is f(x) = |x| a right answer?

Yes. Or a uniformly continuous function which is nowhere differentiable. In fact, given a continuous function $f:I\to\mathbb{R}$ you may construct a continuous function $g:I\to\mathbb{R}$ such that $\|f-g\|_{\infty}<\varepsilon$ for some fixed but arbitrary $\varepsilon>0$ and which is nowhere differentiable. Your answer also shows that given a function $f$ which is differentiable at $x_0$ that the function $f(x)+|x-x_0|$ is not differentiable at $x=x_0$ giving and infinitely weaker form of the aforementioned theorem.

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