JEE Math Practice Question

Let $$f:[0,\infty ) \to [0,\infty )$$ be defined as $$f(x) = \int_0^x {[y]dy} $$ where [x] is the greatest integer less than or equal to x. Which of the following is true?

1. A.f is continuous at every point in $$[0,\infty )$$ and differentiable except at the integer points.
2. B.f is both continuous and differentiable except at the integer points in $$[0,\infty )$$.
3. C.f is continuous everywhere except at the integer points in $$[0,\infty )$$.
4. D.f is differentiable at every point in $$[0,\infty )$$.