JEE Mains · Maths · STD 12 - 5. continuity and differentiation

Let \(f:[0, \infty) \rightarrow[0, \infty)\) be defined as \(\mathrm{f}(\mathrm{x})= \int_{0}^{x}[y] \,d y\) Where \([x]\) is the greatest integer less than or equal to \(x\). Which of the following is true?

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

Verified Solution

Answer & Solution

Correct Answer

(C) \(\mathrm{f}\) is continuous at every point in \([0, \infty)\) and differentiable except at the integer points.

Step-by-step Solution

Detailed explanation

\(\mathrm{f}:[0, \infty) \rightarrow[0, \infty), \mathrm{f}(\mathrm{x})=\int_{0}^{\mathrm{x}}[\mathrm{y}] \mathrm{d} \mathrm{y}\) \(\text { Let } x=n+f, f \in(0,1)\) \(\text { So } f(x)=0+1+2+\ldots+(n-1)+\int_{n}^{n+f} n \,d y\) \(f(x)=\frac{n(n-1)}{2}+n f\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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