Examine the continuity of the function $f(x) = 2x^{2} - 1$ at $x = 3$.

If $f(x) = \begin{cases} \frac{(e^{ax}-1) \log(1+x)}{\sin^2 x} & \text{if } x > 0 \\ 2 & \text{if } x = 0 \\ \frac{\cos 4x - \cos bx}{\tan^2 x} & \text{if } x < 0 \end{cases}$ is continuous at $x = 0$,then $\sqrt{b^2 - a^2} = $

If the function $f(x) = \begin{cases} x + a \sqrt{2} \sin x & \text{if } 0 \leq x \leq \frac{\pi}{4} \\ 2x \cot x + b & \text{if } \frac{\pi}{4} < x \leq \frac{\pi}{2} \\ a \cos 2x - b \sin x & \text{if } \frac{\pi}{2} < x \leq \pi \end{cases}$ is continuous in $[0, \pi]$,then $a - b = $

Discuss the continuity of the function $f,$ where $f$ is defined by $f(x) = \begin{cases} 3, & \text{if } 0 \le x \le 1 \\ 4, & \text{if } 1 < x < 3 \\ 5, & \text{if } 3 \le x \le 10 \end{cases}$ at $x=3.$

If $f(x) = \frac{e^{2x} - (1 + 4x)^{1/2}}{\ln(1 - x^2)}$ for $x \neq 0$,then $f$ has

Let $[t]$ denote the greatest integer $\leq t$ and $\lim_{x \to 0} x[\frac{4}{x}] = A$. Then the function $f(x) = [x^2] \sin(\pi x)$ is discontinuous when $x$ is equal to