Question

How do you prove that the function `f(x)=(x-5)/ |x-5|` is continuous everywhere but x=5?

Answer 1

Use the definition of `abs u = { (-u,"if",u < 0),(u,"if",u >= 0) :}` to rewrite `f` as a piecewise function.

Explanation:

`f(x) = { (-1,"if",x<5),(1,"if",x>5) :}`

So, `f` is constant on `(-oo,5)`, hence continuous on `(-oo,5)`

and `f` is constant on `(5,oo)`, hence continuous on `(5,oo)`.

`f(5)` is not defined, so `f` is not continuous at `5`.