Prove that if $f$ is a rational function then it can not have a jump discontinuity.

Question

Prove that if $f$ is a rational function then it can not have a jump discontinuity.

I have seen many questions similar(but necessarily not exactly the same) to that question here but I still can not figure what are the rigor steps of this proof.

Could anyone give me a hint please?

Answer 1

A jump discontinuity is a point where the function has finite limits from the left as well as from the right and these two are not equal. If $f$ is a rational function it is a ratio of two polynomials say $f=\frac p q$. At any point $x$ where $q(x) \neq 0$ the function is continuous. At point where $q(x)=0$ and $p(x) \neq 0$ the function does not have finite limits from left or right. The case when $p(x)=0=q(x)$ can be avoided by cancelling out any common factors between $p$ and $q$.

Answer 2

Proof: A rational function cannot have a jump discontinuity because it is continuous at every point of its domain (since it is the quotient of two continuous functions).