Limits, Continuity & Differentiability
Continuity And Differentiability
nta_abhyas_2025
Grade 12
Question:
The function $f(x) = \max\{(1-x), (1+x), 2\}$ $\forall x \in \mathbb{R}$ is
discontinuous at exactly two points
differentiable ∀ x ∈ ℝ
differentiable ∀ x ∈ ℝ - {-1, 1}
continuous ∀ x ∈ ℝ - [-1, 1]
Step-by-Step Solution
Key Concept: A piecewise function is differentiable at boundary points only if both continuity and equal left/right derivatives hold
Check continuity at $x = -1$: $f(-1) = 1-(-1) = 2$ and $\lim_{x \to -1^+} f(x) = 2$, so continuous. Check continuity at $x = 1$: $f(1^-) = 2$ and $\lim_{x \to 1^+} f(x) = 1 + 1 = 2$, so continuous. At $x = -1$: $f'(-1^-) = -1$ but $f'(-1^+) = 0$, so not differentiable. At $x = 1$: $f'(1^-) = 0$ but $f'(1^+) = 1$, so not differentiable. Thus the function is differentiable everywhere except at $x = -1$ and $x = 1$.
Correct Answer: C