Let f : [0, ∞) → [0, ∞) be defined as x 0 f(x) = ∫^x_0 [x]dy where [x] is the greatest integer less than or equal to x.

Question

Let f : [0, ∞) → [0, ∞) be defined as x 0 

f(x) = \(\int\limits_0^x [x]dy\)  

where [x] is the greatest integer less than or equal to x. Which of the following is true? 

(1) f is continuous at every point in [0, ∞) and differentiable except at the integer points. 

(2) f is both continuous and differentiable except at the integer points in [0, ∞).

(3) f is continuous everywhere except at the integer points in [0, ∞). 

(4) f is differentiable at every point in [0, ∞).

Answer 1

Correct option (1) f is continuous at every point in [0, ∞) and differentiable except at the integer points. 

ƒ : [0, ∞) → [0, ∞), ƒ(x) = \(\int\limits_0^x [y]dy\)

Let x = n + ƒ, ƒ ∈ (0, 1)

f(x) = \(\frac{n(n-1)}{2}\) + nf

= \(\frac{[\text{x}]([x]-1)}{2}\) + [x]{x}

= \(\frac{n(n-1)}{2}\)

f(x) = \(\frac{n(n-1)}{2}\) (n ∈ N₀)

so ƒ(x) is cont. ∀ x ≥ 0 and diff. except at integer points.