The modulus of x, that is, |x| is a continuous function. In this post, we will learn how to prove mod x is continuous.

Let us now show that mod x is continuous at the point x=0.

## f(x)=|x| is continuous at 0

Let f(x)=|x|.

Note that we have

- $f(0)=|0|=0$
- $\lim\limits_{x \to 0}f(x)=\lim\limits_{x \to 0}|x|=0$

So we obtain that $\lim\limits_{x \to 0} f(x)=f(0)$. So by the definition of continuity, we can say that f(x)=|x| is continuous at x=0.

## f(x)=|x| is continuous at every real number

Note that for a real number c, we have lim_{x→c }f(x) = f(c). So we obtain that the function

f(x)=|x|

is continuous at x=c.

Since c is an arbitrary real number, we conclude that f(x)=|x| is continuous at every real number.

**Note** **that **

- f(x)=|x-a| is continuous at the point x=a.
- A function f(x) is continuous at x=a, then its modulus |f(x)| is also continuous at x=a. This follows from the definition of continuity.

**ALSO READ:**

**|x-a| is continuous at x=a but NOT differentiable**

## FAQs

**Q1: Is |x| continuous at x=0?**

Answer: Mod x, that is, |x| is continuous at x=0. This follows because lim_{x→0 }|x| = 0 = |0|.