# Find points of discontinuity of $\log\left|{\frac{x+2}{x+3}}\right|$

I want to find the points of discontinuity of the following function:

$$f(x)=\log\left|{\frac{x+2}{x+3}}\right|$$

This is defined for $x\neq-2$ and $x\neq-3$. I proceed to find the first derivative:

$$f'(x)=\frac{1}{x^2+5x+6}$$

This is as well defined for $x\neq-2$ and $x\neq-3$. Since $f(x)$ is not defined for these points either, they should not be critical points. Therefore there should be no critical points. However, my textbook says "$x=-2$, $x=-3$ are points of discontinuity for $f$". Any hints on what I'm doing wrong?

• It's a matter of definitions/conventions. I would not say the functions is discontinuous in $x=-2$ or $x=-3$ because the function doesn't even exist there (the points are not in the function's domain), but perhaps your textbook uses another convention. Aug 29, 2018 at 15:12
• critical points are not the same as points of discontinuity. There is no need to calculate the derivative of a function in this case.
– 5xum
Aug 29, 2018 at 15:16
• @5xum thanks, points of discontinuity are "removeable" whereas critical points aren't? Aug 29, 2018 at 15:22
• @Cesare Critical points are minimums and maximums. For example, $0$ is the critical point of $f(x)=x^2$. There is nothing "discontinuous" at $x=0$ for that function.
– 5xum
Aug 29, 2018 at 15:24
• @Cesare On the other hand, for $f'(x)=\sqrt{|x|}$, the derivative at $x=0$ is not defined, but $x=0$ is still a critical point.
– 5xum
Aug 29, 2018 at 15:27