Take the function $f_1(x) = x^{0.5}$. This function is continuous in $\mathbb{R}$ but its derivative $f_1'(x) = 0.5 x^{-0.5}$ is not. If we then consider $f_2(x) = x^{0.9}$, this function is also continuous in $\mathbb{R}$ but with a derivative $f_2'(x) = 0.9 x^{-0.1}$ that is not continuous, but it is 'almost' continuous. Now increasing the exponent some more we have $f_3(x) = x$ which is continuous with continuous derivative, and in fact it is smooth. So increasing the exponent seems to make the functions more regular.
However if we increase the exponent another bit giving us $f_4(x) = x^{1.01}$ we have a continuous function with continuous derivative $f_4'(x) = 1.01 x^{0.1}$. But it can only be differentiated once..so even though $f_4$ has a higher exponent than $f_3$ it is not as regular! But then if we increase the exponent to the next highest integer we get $f_5(x) = x^2$ and this is again infinitely differentiable like $f_3$!
- So it seems increasing the exponent does make functions more regular, but the regularity 'resets' at every integer exponent? Is this how I should think of regularity of functions?
- Can this phenomenon be explained by Holder spaces (which I know little about)? What Holder spaces would $f_1, f_2, f_3, f_4$, and $f_5$ fall into?
- The functions I used here are very simple, how is regularity of functions handled in the case of more complicated functions which could involve multiple terms and summations?