Let [f:[0, ) [0, )] be defined as [f(x) = _0^x [y]dy ] where [x] is the greatest integer l…

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?

Accepted Answer

$$f:[0,\infty ) \to [0,\infty ),f(x) = \int_0^x {[y]dy} $$

Let $$x = n + f,f \in (0,1)$$

So, $$f(x) = 0 + 1 + 2 + ... + (n - 1) + \int\limits_n^{n + f} {n\,dy} $$

$$f(x) = {{n(n - 1)} \over 2} + nf$$

$$ = {{[x]([x] - 1)} \over 2} + [x]\{ x\} $$

Note $$\mathop {\lim }\limits_{x \to {n^ + }} f(x) = {{n(n - 1)} \over 2},\mathop {\lim }\limits_{x \to {n^ - }} f(x) = {{(n - 1)(n - 2)} \over 2} + (n - 1)$$

$$ = {{n(n - 1)} \over 2}$$

$$f(x) = {{n(n - 1)} \over 2}(n \in {N_0})$$

so f(x) is cont. $$\forall$$ x $$\ge$$ 0 nd diff. except at integer points

Solutions · 1