Limits, Continuity & Differentiability
Continuity of Combined Floor and Fractional Part Functions
nta_pyq_2023_apr
Grade 12
Question:
Let $f(x)=[x^2-x]+|-x+[x]|$, where $x\in\mathbb{R}$ and $[t]$ denotes the greatest integer less than or equal to $t$. Then, $f$ is
continuous at x=0, but not continuous at x=1
continuous at x=1, but not continuous at x=0
continuous at x=0 and x=1
not continuous at x=0 and x=1
Step-by-Step Solution
Key Concept: $|-x+[x]|=|-(x-[x])|=\{x\}$ (fractional part). So $f(x)=[x^2-x]+\{x\}$.
Continuous at $x=1$, discontinuous at $x=0$.
Correct Answer: 2