Limits, Continuity & Differentiability Question - None

Question

Let $f:\mathbb{R}\to\mathbb{R}$ be differentiable with $|f(x)-f(y)|\le|x-y|^3$ for all $x,y\in\mathbb{R}$ . If $$f(10)=100$$ , then $$f(20)=$$

Accepted Answer

Step 1: Put $|f(x)-f(y)|\le|x-y|^3$ : $x,y\in\mathbb{R}$

Step 2: So $$f(20)=$$ for all $$x$$ , meaning $$f$$ is constant.

Step 3: $f(10)=100\implies f(x)=100$ for all $$x$$ . So $$f(20)=100$$ .

Answer: (D) 100

Trap: The exponent 3 on |x-y| is key — it forces f'=0. With exponent 1 this would not work.

Key Concept: Lipschitz condition with exponent >1 forces derivative to be 0 → constant function

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