List of top Mathematics Questions on Continuity and differentiability asked in JEE Advanced

For a real number \(x\), let \([x]\) denote the greatest integer less than or equal to \(x\). For a finite set \(S\), let \(|S|\) denote the number of elements in the set \(S\). Consider the functions \[ f : (-3,3) \to (-\infty,\infty) \] and \[ g : (-3,3) \to (-\infty,\infty) \] defined by \[ f(x) = [x^3]\ln\!\left(1 + \sin^2\!\left(\pi(x-[x])\right)\right) \] and \[ g(x) = x^3 \sin^2\!\left(\pi \ln(1 + x - [x])\right). \] Let \[ A = \{x \in (-3,3) : f \text{ is discontinuous at } x\} \] and \[ B = \{x \in (-3,3) : g \text{ is discontinuous at } x\}. \] Then the value of \[ |A| + 2|B| - |A \cap B| \] is:
Consider the function \[ f:\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\to(-\infty,\infty) \] defined by \[ f(x)=\left(|x|+|x-1|\right)\sin x+[x\sin x] \] where \([x\sin x]\) denotes the greatest integer less than or equal to \(x\sin x\). Let \(\alpha\) be the total number of points in the interval \[ \left(-\frac{\pi}{2},\frac{\pi}{2}\right) \] at which \(f\) is NOT continuous, and let \(\beta\) be the total number of points in the interval \[ \left(-\frac{\pi}{2},\frac{\pi}{2}\right) \] at which \(f\) is NOT differentiable. Then the value of \[ \alpha+\beta \] is ________.
Let \(\mathbb R\) denote the set of all real numbers. Let \[ f:\mathbb R\to\mathbb R \] be an arbitrary function and let \[ g:\mathbb R\to\mathbb R \] be the function defined by \[ g(x)=xf(x),\qquad \forall x\in\mathbb R \] Then which of the following statements is (are) TRUE?