show that f(x) = |x| is a continuous function everywhere - Maths - Continuity and Differentiability

Question

show that f(x) = |x| is a continuous function everywhere?

Gursheen kaur. · Accepted Answer

Given: f(x) = |x|

According to the theorem, the modulus function is everywhere continuous.

Also, the identity function is everywhere continuous.

⇒If f is continuous, then |f| is continuous.

∴ f(x) = |x|is everywhere continuous.

To Prove: Let |x| = x , if x ≥0 and -x, if x< 0

Since |x| = x for all x > 0, we find the following right hand limit.

$\lim_{x \to 0^{+}} | x | = \lim_{x \to 0^{+}} x = 0$

Now, |x| = -x for all x < 0, we find the following left hand limit.

$\lim_{x \to 0^{-}} | x | = \lim_{x \to 0^{-}} - x = 0$
∴The two one-sided limits agree, the two sided limit exists and is equal to 0, and we find that the following is true.

$\lim_{x \to a} | x | = 0 = | a |$

⇒ For the absolute value function to be continuous at a = 0.

We remark that it is even less difficult to show that the absolute value function is continuous at other (nonzero) points in its domain. (There is no need to consider separate one sided limits at other domain points).

In fact, the absolute value function is continuous (on its entire domain R).

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