A rational function is a function f of the form pq where p and q are polynomial functions. The domain of f is {x∈R:q(x)≠0}. Prove every rational function is continuous. I have previously proved that every polynomial function p(x)=a0+a1x+…+…anxn is continuous on R. Could I use a theorem that states fg is continuous at x0 if g(x0)≠0? Then p(x) (which is already proved to be continuous) over q(x) would be continuous?

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A rational function is a function f of the form pq where p and q are polynomial functions. The domain of f is {x∈R:q(x)≠0}. Prove every rational function is continuous.  I have previously proved that every polynomial function p(x)=a0+a1x+…+…anxn is continuous on R. Could I use a theorem that states fg is continuous at x0 if g(x0)≠0? Then p(x) (which is already proved to be continuous) over q(x) would be continuous?

Cheryl Stark · Accepted Answer

A function f of x is continuous at x=a if and only if  limx→af(x)=f(a)  Obviously, as you’ve stated, not every rational function is continuous on the entire real line, so you just need to prove there is some interval I⊂R on which f is continuous—that is, there is an a∈I that satisfies the above criterion.  So argue that there is at least one a∈R{x:q(x)=0} such that  limx→ap(x)q(x)=p(a)q(a)  limx→ap(x)limx→aq(x)=p(a)q(a)  limx→ap(x)limx→aq(x)∈R  Since there is only a finite number of x:q(x)=0 and since p(x) is defined across the whole real line, then there are an infinite number of valid a.

A rational function is a function f of the form \frac{p}{q} where p and q are polynomial funct

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