Question

Let $f(x)=[x]$, $x\in(0,6)$, where $[x]$ is the greatest integer function. Then the number of discontinuities of $f(x)$ is:

• A. 1
• B. 2
• C. 3
• D. 4
• E. 5

Hint:

For an open interval $(a, b)$, the number of discontinuities of $[x]$ is $\lfloor b - \epsilon \rfloor - \lfloor a \rfloor$, effectively counting integers between $a$ and $b$.

Answer 1

Answer: (E)

Step 1: Concept
The greatest integer function $[x]$ is discontinuous at every integer point.

Step 2: Analysis
The interval given is $0 < x < 6$. The integers within this open interval are $\{1, 2, 3, 4, 5\}$.

Step 3: Conclusion
There are 5 such integer points where the function jumps. Hence, there are 5 points of discontinuity. Final Answer: (E)