How do I find all points where the following function is discontinuous?

Question

f(x) = 1/(5-(x^2+9)^1/2)((x^2+1)/((1-x)^4))^1/2

Answer 1

A function is said to have a point of discontinuity at x = a or the graph of the function has a gap at x = a, if the original function is undefined for x = a, whereas the related rational expression of the function in simplest form is defined for x = a.
... it looks like the denominators are (5 - √(x² + 9)) and √(1-x)⁴ then function is undefined if

1. 5 - √(x² + 9) = 0       or        2. √(1-x)⁴ = 0
     √(x² + 9) = 5                               x = 1
         x² + 9 = 5²  
         x² = 16
         x = ± 4  
The points of discontinuity are {- 4, 1, +4}

Answer 2

You want to consider where the denominator will equal zero.

There are two pieces to check.

(1-x) = 0               So, x = 1

5-sqrt(x^2+9)=0   Or, x= +-5

Graphing it can also help.