Question

Examine the following functions for continuity :

$f\left(x\right)=\frac{1}{x-5},x\ne 5$

Answer 1

$f\left(x\right)=\frac{1}{x-5}$ is a quotient functions of two polynomial functions, so f(x) is continuous for all values of x provided $x\ne 5$.

Note In any rational function $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ and if P(x) and q(x) are polynomials and q(x) is zero at any value of x and p(x) = 0, then f(x) is not continuous at that point.