Let $f:(0,1) \rightarrow \mathbb{R}$ be the function defined as $f(x)=[4 x]\left(x-\frac{1}{4}\right)^2\left(x-\frac{1}{2}\right)^2$, where $[x]$ denotes the greatest integer less than or equal to $x$. Then which of the following statements is(are) true?
Select ALL correct options:
A
The function $f$ is discontinuous exactly at one point in $(0,1)$
B
There is exactly one point in $(0,1)$ at which the function $f$ is continuous but NOT differentiable
C
The function $f$ is NOT differentiable at more than three points in $(0,1)$
D
The minimum value of the function $f$ is $\frac{1}{512}$
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