Question

The value of \(f(x) = x + |x|\) is continuous for:

• A. \(x \in (-\infty, \infty)\)
• B. \(x \in (-\infty, \infty) - \{0\}\)
• C. Only \(x>0\)
• D. No value of \(x\)

Hint:

Functions involving (|x|) are usually continuous everywhere, but may fail to be differentiable at points where the expression changes form.

Answer 1

The Correct Option is A

Step 1: Write the function in piecewise form. \[ f(x) = x + |x| = \begin{cases} x + x = 2x, & x \ge 0
x - x = 0, & x<0 \end{cases} \] Step 2: Check continuity at the critical point \(x = 0\).

Left hand limit: \[ \lim_{x \to 0^-} f(x) = 0 \]
Right hand limit: \[ \lim_{x \to 0^+} f(x) = 2(0) = 0 \]
Value of the function at \(x=0\): \[ f(0) = 0 + |0| = 0 \]
Step 3: Compare LHL, RHL and \(f(0)\). \[ \text{LHL} = \text{RHL} = f(0) \] Hence, \(f(x)\) is continuous at \(x=0\). Step 4: Check continuity elsewhere. Both expressions \(2x\) and \(0\) are polynomials (or constants), which are continuous for all real \(x\). Step 5: Final conclusion. \[ f(x) \text{ is continuous for all } x \in (-\infty, \infty) \]