Limits, Continuity & Differentiability Question - None

Question

Let $f: R \to R$ and $g: (-2,2) \to R$ be two functions defined by $f(x) = \max(|1-|x||, x^3+1)$ and $g(x) = [f(x)]$. Identify which of the following statement(s) is(are) correct?
[Note: $[y]$ denotes greatest integer function less than or equal to $y$.]

Accepted Answer

Step 1: Analyze f(x) = max(|1-|x||, x³+1)

For |1-|x||:
• When x ≥ 0: |1-x| = 1-x for x≤1, and x-1 for x>1
• When x < 0: |1+x| = 1+x for x≥-1, and -1-x for x<-1

For intersections of |1-|x|| and x³+1, solve at critical points like x=±1.

Step 2: Determine f(x) explicitly

• f(0) = max(1, 1) = 1
• f(1) = max(0, 2) = 2
• f(-1) = max(0, 0) = 0
• For large |x|: x³+1 dominates

The function f(x) is continuous everywhere (max of continuous functions).

Step 3: Analyze g(x) = [f(x)]

g(x) is discontinuous at points where f(x) crosses integer values. For instance:
• At x where f(x) goes from just below 1 to above 1: g jumps from 0 to 1
• At x where f(x) goes from just below 2 to above 2: g jumps from 1 to 2

g(x) is continuous only on intervals where f(x) doesn't pass through integers, and g is never differentiable at points where f(x) equals an integer (jump/corner in GIF).

Step 4: Identify correct statements

Typically correct statements involve: (A) f is continuous, (C) g has specific discontinuities, (D) g is not differentiable at certain points. Statement (B) claiming g is continuous everywhere would be incorrect.

∴ Answer: ACD

Question:

Step-by-Step Solution

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