Continuity and Differentiability Questions (9)

Let $[\cdot]$ denote the greatest integer function, and let $f(x)=\min\{\sqrt{2}x,\,x^2\}$. Let $S=\{x\in(-2,2): g(x)=|x|\,[x^2]$ is discontinuous at $x\}$. Then $\displaystyle\sum_{x\in S}f(x)$ equals

If $f(x) = \begin{cases} \dfrac{a|x|+x^2-2\sin|x|\cos|x|}{x} & x\neq0 \\ b & x=0 \end{cases}$ is continuous at $x=0$, then $a+b$ is equal to

Let $[t]$ denote the greatest integer less than or equal to $t$. If the function $f(x) = \begin{cases} b^2\sin\!\left(\dfrac{\pi}{2}\!\left[\dfrac{\pi}{2}(\cos x+\sin x)\cos x\right]\right) & x < 0 \\[6pt] \dfrac{\sin x - \tfrac{1}{2}\sin 2x}{x^3} & x > 0 \\[4pt] a & x = 0 \end{cases}$ is continuous at $x=0$, then $a^2+b^2$ is equal to

{ ⎪ ⎪ ⎪$(1 + ax)$$1/x$,$x < 0$$1 + b$,$x = 0$Let f$(x) = 1/2$$(x+4) -2$⎪ ⎪ ⎪ ,$1/3$$(x+c) -2$be continuous at$x = 0.$Then e bc is equal to a

The number of points of discontinuity of the function f (x) = [ 2 x 2 ] - [$\sqrtx$], x$\ in $[0, 4] , where [$\cdot$] denotes the greatest integer function is ________

If the function $f(x)=\dfrac{e^{x}\!\left(e^{\tan x-x}-1\right)+\log_e(\sec x+\tan x)-x}{\tan x-x}$ is continuous at $x=0$, then the value of $f(0)$ is equal to

Let $f(x) = \begin{cases} \dfrac{ax^2+2ax+3}{4x^2+4x-3} & x \neq -\dfrac{3}{2},\,\dfrac{1}{2} \\ b & x = -\dfrac{3}{2},\,\dfrac{1}{2} \end{cases}$ be continuous at $x=-\dfrac{3}{2}$. If $f\circ f(x)=\dfrac{7}{5}$, then $x$ is equal to:

Let f : R$\to$R be a twice differentiable function such that (sin x cos y)(f$(2x + 2y) - f$$(2x - 2y)) = (cos$x sin y)(f$(2x + 2y) + f$$(2x - 2y))$, for all x, y$\ in $R. If f (0) = ′ 1 2 , then the value of 24f ′′ ( 5$\pi$3 ) is:

Let $\alpha,\beta\in\mathbb{R}$ be such that the function $f(x)=\begin{cases}2\alpha(x^2-2)+2\beta x & x<1\\(\alpha+3)x+(\alpha-\beta) & x\geq1\end{cases}$ be differentiable at all $x\in\mathbb{R}$. Then $34(\alpha+\beta)$ is equal to