Question

The function $f\left(x\right)=|x|+\frac{|x|}{x}$ is

• A. Discontinuous at origin because $|x|$ is discontinuous there
• B. Continuous at origin
• C. Discontinuous at origin because both $|x|$ and $\frac{|x|}{x}$ are discontinuous there
• D. Discontinuous at the origin because $\frac{|x|}{x}$ is discontinuous there.

Answer 1

Answer: (D)

Discontinuous at the origin because $\frac{|x|}{x}$ is discontinuous there.

$f\left(x\right)=|x|+\frac{|x|}{x}$

Left Hand Limit:

$\underset{x\to 0^{-}}{lim}|x|+\frac{|x|}{x}=-x-1$

Right Hand Limit:

$\underset{x\to 0^{+}}{lim}|x|+\frac{|x|}{x}=x+1$

Given function $f\left(x\right)$ is the combination of two functions.

If both functions are continuous at origin then the function $f\left(x\right)$ will be continuous at origin.

But here $|x|$ is continuous at origin, and $\frac{|x|}{x}$ is discontinuous at origin.

Therefore, $f\left(x\right)$ will also be discontinuous at origin due to discontinuity of $\frac{|x|}{x}$ at origin.