Rational Function is Continuous

Theorem

A rational function is continuous at every point at which it is defined.

Thus a rational function is continuous on every interval of $\R$ not containing a root of the denominator of the function.

Proof

Let $\displaystyle R \left({x}\right) = \frac {P \left({x}\right)} {Q \left({x}\right)}$ be a rational function, defined at all points at which $Q \left({x}\right) \ne 0$.

Let $c \in \R$.

From Polynomial is Continuous‎:

$\displaystyle \lim_{x \to c} \ P \left({x}\right) = P \left({c}\right)$

and:

$\displaystyle \lim_{x \to c} \ Q \left({x}\right) = Q \left({c}\right)$

Thus by Quotient Rule for Limits of Functions:

$\displaystyle \lim_{x \to c} \ R \left({x}\right) = \lim_{x \to c} \ \frac {P \left({x}\right)}{Q \left({x}\right)} = \frac {P \left({c}\right)}{Q \left({c}\right)}$

whenever $Q \left({c}\right) \ne 0$.

So whatever value we choose for $c$ such that $Q \left({c}\right) \ne 0$, we have that $R \left({x}\right)$ is continuous at $c$.

From the definition of continuity on an interval, the second assertion follows.

$\blacksquare$

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 8.13$