Question

How to prove that `f(x):=|x|/x`, if `x!=0` and `f(x)=0`, if `x=0`, is not continuous at 0 ?

,

Answer 1

Please see below.

Explanation:

Recall that `absx = {(x,x >= 0),(-x,x < 0):}`

Therefore, `absx/x = {(x/x=1,x > 0),(-x/x=-1,x < 0):}`

Hence,

`lim_(xrarr0^+)f(x) = 1` and `lim_(xrarr0^-)f(x) = -1` .

Because the one-sided limits are not equal, the limit at 0 does not exist.

Therefore `f` is not continuous at `0`.