$$ p(\theta) = \left[\lim_{t \to0} p(t) \right] \lim_{n \to \infty}\prod_{i=0}^{n}( \frac{\cos^2 \frac{ \theta}{2^i} +1}{2})$$
I know each term in the product is differentiable but does that mean the total product will be ? I thought of doing a repeated product rule, but I was a bit worried that maybe due to the limit, the differentiability would be effected.
The answer is No, it need not even be continuous. For a simple example, consider the infinite product $\prod_{n=1}^\infty g_n(x)$ where $$ g_n(x) = \frac{f_{n+1}(x)}{f_n(x)}, \quad f_n(x) = \frac{e^{nx}}{1 + e^{nx}} \, . $$ Thus clearly $\prod_{n=1}^\infty g_n(x) = \lim_{n \to \infty} f_n(x)$ which jumps from $y = 0$ for $x < 0$ to $y = 1$ for $x > 0$.
