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Are there any "smooth-looking" functions that are not infinitely differentiable? By "smooth-looking", I refer to the nature of the plotted graph of the function. Thanks.

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    One could conceivably construct a piecewise function such that its values and derivatives up to a certain order agree at certain points. For instance, splines are constructed to be $C^2$ (that is, the second derivative is continuous, but not the third derivative).2011-09-21

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The function $f(x)=x^{\frac{1}{3}}$ is not even $C^1$. (It's an auto-homeomorphism of $\mathbb{R}$, though!)

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    I believe it is well-defined and $C^\infty$ for all nonzero real numbers. Every real number has a real cube root and away from zero $f'(x) = \frac{1}{3}x^{-\frac{2}{3}}$, $f''(x)=\frac{1}{3}\frac{-2}{3}x^{-\frac{5}{3}}$, etc.2011-09-21