Question

Which of the following functions is not continuous on the set of real numbers?

• A. \(f(x)=|x|+|x+1|+|x-2|\)
• B. \(f(x)=|\cos x|\)
• C. \(f(x)=x^3+|x|\)
• D. \(f(x)=[x]\)

Hint:

The greatest integer function \[ [x] \] has jump discontinuities at every integer. Absolute value functions remain continuous whenever the original function is continuous.

Answer 1

The Correct Option is D

Concept:

• Sum of continuous functions is continuous.

• Absolute value of a continuous function is continuous.

• Greatest integer function \([x]\) has jump discontinuities at every integer.

Step 1: Examine each option. \[ \begin{aligned} |x|+|x+1|+|x-2| &\rightarrow \text{Continuous} \\ |\cos x| &\rightarrow \text{Continuous} \\ x^3+|x| &\rightarrow \text{Continuous} \\ [x] &\rightarrow \text{Discontinuous at integers} \end{aligned} \]

Step 2: Consider the greatest integer function. At any integer \(n\), \[\begin{aligned} \lim_{x\to n^-}[x] &=n-1 \\ \lim_{x\to n^+}[x] &=n \end{aligned}\] Since the left-hand and right-hand limits are different, \[ [x] \] is not continuous at integers. \[\begin{aligned} \boxed{f(x)=[x]} \end{aligned}\] Hence, option \(\mathbf{(D)}\) is correct.