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Hi could you help me with the following:

If I have a function $g \in C^2(0,R)$ with $|g''(x)| \le M$, i.e. its second derivative is bounded except a finite number of points where at those irregularity points none of the assumptions mentioned above holds can you give me a sequence of functions $f_n$ with $f_n^\prime \longrightarrow g'$ and $f_n \longrightarrow g$ uniformly and $|f_n^{\prime\prime}(x)| \le M$?

I think about polynomials but can not justify that they have all the requirements??

Thanks a lot!!

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    What kind of functions? $f_n = g$ for all $n$ is a trivial example. – 2012-10-08
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    I am sorry i should have added g∈C2(0,R) except finite points x_1, x_2, .. ,x_n. The question is how to find a sequence f_n which are completely nice with those properties? Thank you – 2012-10-08
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    You can (and should) edit your post to add this information. Also, what kind of regularity does the function at these points have? Is it still continuous and differentiable? Is the derivative continuous there? – 2012-10-08

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