$f(x)$ is defined as follows and $g(x)=|f(x)|$. I have to find the number of points of non-differentiability of $g(x)$, which will be much easier once I have dealt with this product and simplified $f(x)$.
$$f(x)=\lim_{n\to \infty}\prod_{k=1}^{n}\left(\frac{1+2\cos(2x/3^k)}{3}\right)$$
Any ideas. Thanks.
$$1+2\cos2A=1+2(1-2\sin^2A)=\dfrac{\sin3A}{\sin A}\text{ if }\sin A\ne0$$
$$\prod_{k=1}^n\left(1+2\cos\dfrac{2x}{3^k}\right)=\prod_{k=1}^n\dfrac{\sin\dfrac x{3^{k-1}}}{\sin\dfrac x{3^k}}=\dfrac{\sin x}{\sin\dfrac x{3^n}}\text{ by Telescoping Series }$$
Finally $\lim_{h\to0}\dfrac{\sin h}h=1$
Can you identify $h$ here?