CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

For f to be defined,

x - 5 $\neq$ 0 i.e. x $\neq$ 5

$\therefore$ D_f= Set of real number except 5 = R = -{5}

Let c $\neq$ 5 be any real number.

Also $^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{c}}\text{f(x)}-^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{c}}\Big(\frac{1}{\text x- 5}\Big)= \frac{1}{\text{c}-5}$

$\therefore\ ^{\ \ \text{Lt}}_{\text{x}\rightarrow\text{c}}\text{f(x)} = \text{f(c)}$

$\therefore$ f is continuous at x = c.

But c $\neq$ 5 is any real number

$\therefore$ f is continuous at every real number $\text{c}\in \text{D}$

$\therefore$ f is continuous function.