# Mod x is continuous: Proof

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

1. $f(0)=|0|=0$
2. $\lim\limits_{x \to 0}f(x)=\lim\limits_{x \to 0}|x|=0$

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 limxc 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.

Sinx is continuous: Proof

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

Derivative of mod x

## FAQs

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

Answer: Mod x, that is, |x| is continuous at x=0. This follows because limx0 |x| = 0 = |0|.