Question

Let \(f:\mathbb R\to\mathbb R\) be a function such that \(f(x+y)=f(x)+f(y)\) for all \(x,y\in\mathbb R\). If \(f(x)\) is differentiable at \(x=0\), then which one of the following is incorrect?

• A. \(f(x)\) is continuous, \(\forall x\in\mathbb R\)
• B. \(f(x)\) is constant, \(\forall x\in\mathbb R\)
• C. \(f(x)\) is differentiable, \(\forall x\in\mathbb R\)
• D. \(f(x)\) is differentiable only on a finite interval containing zero

Hint:

Cauchy equation + differentiability ⇒ linear.

Answer 1

The Correct Option is B

Step 1: Additive + differentiable at 0 implies linear: \[ f(x)=kx \]
Step 2: Hence \(f(x)\) is continuous and differentiable everywhere.
Step 3: It is not necessarily constant.