Question

What is the discontinuity of the function `f(x) = |x-5|/(x-5)` ?

Answer 1

There is a jump discontinuity at `x=5`.

Explanation:

The signum function `sgn(x)=|x|/x` returns the sign of a number. For example, `sgn(-6)=-1` because the input was a negative number. The signum function described above has a jump discontinuity at 0 because a negative input returns -1 and a positive input returns +1. Therefore, `lim_(xrarr0^-)|x|/x=-1` and `lim_(xrarr0^+)|x|/x=+1`. The function `f(x)=|x-5|/(x-5)` is just the signum function shifted right 5 units, to the jump discontinuity of the signum function at `x=0` is also shifted right 5 units to `x=5`, so there is a jump discontinuity at `x=5`.