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?

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?

f is continuous at every point in $[0,\infty )$ and differentiable except at the integer points.

f is both continuous and differentiable except at the integer points in $[0,\infty )$.

f is continuous everywhere except at the integer points in $[0,\infty )$.

f is differentiable at every point in $[0,\infty )$.

Solution

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