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

Let $$c \in \R$$.

From Polynomial is Continuous‎, $$\lim_{x \to c} P \left({x}\right) = P \left({c}\right)$$ and $$\lim_{x \to c} Q \left({x}\right) = Q \left({c}\right)$$.

Thus by Combination Theorem for Functions: Quotient of Limits, $$\lim_{x \to c} R \left({x}\right) = \lim_{x \to c} \frac {P \left({x}\right)}{Q \left({x}\right)} = \frac {P \left({c}\right)}{Q \left({c}\right)}$$.

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.