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Question:

Give an example of a sequence of continuously differentiable function $(f_n)$ on $[0,1]$ so that $f_n \to f$ uniformly, but $f$ is not differentiable at all points of $[0,1]$.

My Thoughts:

Would a Fourier Series be a correct answer to this question? Take the Triangle Wave for instance. Wikipedia gives me the following equation: enter image description here

Here $\omega$ is the angular frequency. Instead of $\infty$ in the sum, could each of my $f_n$ be $\sum_{k=0}^n$. In the limit, this sum of continuously differentiable functions converges to a function that is not differentiable at its cusps.

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    @Joachim: Perhaps you were thinking of analytic (or holomorphic, if you prefer) functions? Uniform limits of analytic functions are analytic.2012-08-24

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A simpler alternative to the triangular wave is $f_n(x)=\sqrt{(x-\frac12)^2+\frac1n}$ and $f(x)=|x-\frac12|$.