For what $x\in[0,1]$ will the function $y = \sum\limits_{k = 1}^\infty\frac{\sin( k!^2x )}{k!}$ be differentiable? How do you know?
Here is the equation expressed more clearly on Wolfram Alpha. The only difference is that 10 should be Infinity (Wolfram apparently can't handle that yet).
I'm trying to understand for what $x\in[0,1]$ this function is differentiable. I've used a computer to plot the graph of y' (the derivative of the function) with the upper limit (top number of sigma) as 10, and then with the upper limit as 11, 12... it looks like these "zig-zags" continue to exist as you "go deeper" into the function.
...so I'm thinking the values of $x\in[0,1]$ that make the function differentiable are all of them... But is my line of thinking correct? How can I validate?