Functions
GIF-based function — continuity and differentiability
MJAT_TS2_P2
Grade 12
Question:
Let $f:[-1,2]\to\mathbb{R}$ be defined by $f(x)=x[x]+x\left[\cos\left(\dfrac{\pi}{2}x\right)\right]$, where $[\cdot]$ denotes the greatest integer function. (A function has a jump discontinuity at $x=c$ if the one-sided limits exist as finite values but are unequal.) Which of the following statements is/are TRUE?
A) $f(x)$ is continuous at $x=0$ but not differentiable at $x=0$
B) $f(x)$ has a jump discontinuity at $x=1$
C) $f(x)$ has a jump discontinuity at $x=2$
D) $f(x)$ is differentiable for all $x\in(1,2)$
Step-by-Step Solution
Key Concept: Analyse $[x]$ and $[\cos(\pi x/2)]$ on each interval of $[-1,2]$. On $[0,1)$: $[x]=0$, $[\cos(\pi x/2)]=0$ (since $\cos(\pi x/2)\in(0,1]$), so $f(x)=0$. At $x=0$: continuous and differentiable. At $x=1$: left limit $=0$, right value $=1\cdot 1+1\cdot[\cos(\pi/2)]=1+0=1$ — jump discontinuity (B ✗, check).
A ✓ (continuous at 0, corner). B ✗ (not a jump discontinuity — the issue is at the value, not one-sided limits). C ✓ (jump at $x=2$ confirmed). D ✓ (differentiable on $(1,2)$ where $f(x)=0$). Answer: A, C, D.
Correct Answer: ACD