Let $f(x) = \begin{cases} \frac{(x - 1)(6x - 1)}{2x - 1}, & \text{if } x \neq \frac{1}{2} \\ 0, & \text{if } x = \frac{1}{2} \end{cases}$. Then at $x = \frac{1}{2}$,

If the function $f(x) = \begin{cases} \frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}} & , x \neq 0 \\ a \ln 2 \ln 3 & , x=0 \end{cases}$ is continuous at $x=0$,then the value of $a^2$ is equal to

Examine the continuity of $f$,where $f$ is defined by $f(x) = \begin{cases} \sin x - \cos x, & \text{if } x \neq 0 \\ -1, & \text{if } x = 0 \end{cases}$

Examine the following function for continuity: $f(x) = x - 5$.

Let $f: R \rightarrow R$ be a function defined as $f(x)=\begin{cases} \frac{\sin (a+1) x+\sin 2 x}{2 x} & , \text{if } x<0 \\ b & , \text{if } x=0 \\ \frac{\sqrt{x+b x^{3}}-\sqrt{x}}{b x^{5 / 2}} & , \text{if } x>0 \end{cases}$. If $f$ is continuous at $x=0$,then the value of $a+b$ is equal to ....... .

If $f: R \rightarrow R$ is defined by $f(x) = \begin{cases} \frac{x + 2}{x^2 + 3 x + 2}, & x \in R - \{-1, -2\} \\ -1, & x = -2 \\ 0, & x = -1 \end{cases}$ then $f$ is continuous on the set