Problem

What is

What is

Is |x| continuous at 0?

Solution

To find the first limit, we note that when x is greater than zero, |x| = x. Therefore the limit as x approaches zero from the right of x is zero.

To find the second limit, we note that when x is less than zero, |x| = –x. Therefore the limit as x approaches zero from the left of –x is zero.

Since the one-sided limits approach the same value, we can drop the restrictions and write

|x| is continuous at zero if the following conditions hold:

1. |x| is defined at zero; that is, |0| is equal to something, call it A;
   
2. the limit of |x| as x approaches 0 exists; call this L;
   
3. A = L.

We already showed condition 2 is satisfied. Condition 1 is easy, since we can calculate the absolute value of zero.

Condition 3 is also satisfied, because the value of |0| is equal to

Therefore |x| is continuous at zero.