Question

Prove that every rational function is a continuous.

Answer 1

Every rational number is of form $=\frac{P}{2}$

where $q\ne 0$ & p, q are polynomial function.

Let $f\left(x\right)=\frac{P\left(x\right)}{q\left(x\right)},q\left(x\right)\ne 0$

where $p\left(x\right)$ & $q\left(x\right)$ are polynomials

Since $p\left(x\right)$ & $q\left(x\right)$ are polynomials are we know that every polynomial function is continuous.

$p\left(x\right),q\left(x\right)$ are continuous functions if p, q are continuous then $\frac{p}{q}$ is continuous. Thus $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ is continuous except at $q\left(x\right)=0$.