Find out where the function
is both continuous and differentiable
As this is a rational function, you know that it is discontinuous where it has vertical asymptotes, which is where its denominator equals 0. That means you have to solve this:
That means the function is continuous for all values of except and . You write this mathematically as (all in except and ).
A function has to be continuous at a given point to be differentiable at that point, so you can conclude that the function is not differentiable at the points and . The question is if there are other points where is not differentiable. You check that by finding out whether
for all . First, you find an expression for the derivative:
You know that is defined for all except where the denominator of is not defined. This happens when the denominator is 0. You set the denominator equal to 0 and solve for :
That means is differentiable for all (all in except and ).