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Let $\{f_n\}$ be a sequence of smooth functions which converges to a function $f$. If the convergence is not uniform at a point $a$ the $f$ is discontinuous at $a$. Is there any different type of convergence where if it happens at $a$ then $f$ is continuous at $a$ but is not differentiable.

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Let $\{f_n\}$ be a sequence of smooth functions which converges to a function $f$.If there is a discontinuity in $f$ at some point $a$ then the convergence is nonuniform.Is there any different type of convergence needed for $f$ to be continuous at some point $a$ but not differentiable ?

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    @Jonas Meyer: yeah got it. when you are translating everything along the $x$-axis towards the boundaries, there is no need to bother about $sup|f_n-f|$.2010-11-21

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P[a,b], the set of polynomials on [a,b] (obviously a subset of the smooth functions on [a,b]) is dense in C[a,b]. In addition the set of functions that are differentiable at (at least) a single point in [a,b] are of the first category in C[a,b].

In some sense "most" convergent sequences of smooth functions converge to nowhere differentiable functions.

Semi-related, but you might in interested in the fact that $C^k(\Omega)$, $\Omega \subset \mathbb{R}^n$ open, can be made into a Fréchet space for any $k = 1,2,\ldots, \infty$. See Wikipedia's article on Fréchet spaces, the topology on $C^k(\Omega)$ being uniform convergence of $f_n$ and all of its derivatives up to order $k$ (q.v. multiindex)

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    I can't be bothered to edit this again so a correcting comment will have to do. The last sentence should say that the usual metric on $C^k(\Omega)$ is the same topology as uniform convergence of $(D^\alpha f_n)$ for all multi-indices $\alpha$ s.t. $|\alpha| \leq k$ *on compact subsets of* $\Omega$.2010-11-20