So,
Given the function:
[tex]f(x)=\frac{4}{x^2-36}[/tex]To check if the function is continuous in the entire number real line, we need to analyze the restrictions in the domain.
As you can notice, the denominator of a rational function can't be zero, so:
[tex]x^2-36\ne0[/tex]We're going to find the values of x such that:
[tex]x^2-36=0[/tex]This is:
[tex]\begin{gathered} x^2=36 \\ x=\pm6 \end{gathered}[/tex]As you can see, "x" can't take the values of 6 and -6. If that happens, the function is not defined. Thus, the function is not continuous on the entire real number line.