Continuity and differentiability at zero of a piecewise oscillating function

Let $\{a_n\}$ be a sequence of positive real numbers such that $\lim_{n\to\infty}a_n^{1/n}=\frac14$. Then the function

$$f(x)=\begin{cases}x\sin\dfrac1{x^2}, & x\ne0,\ \lim_{n\to\infty}\dfrac{\log(1+a_n)}{\sin\left(a_n+\frac\pi2\right)}, & x=0,\end{cases}$$

is