QJEEM21Ma10fd: JEE Main 2021 Mathematics - Class 12 JEEM21Ma10fd Solution: Let be defined as where [x] is the greatest integer less than…

Question

Verified Solution • 2025–26 Session moderate

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?

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

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

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