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The function has derivative - [Iran Second Round 1984]

Source:

December 30, 2010
functioncalculusderivativegeometrygeometric transformationalgebra unsolvedalgebra

Problem Statement

Let f:R→Rf : \mathbb R \to \mathbb R be a function such that f(x+y)=f(x)⋅f(y)∀x,y∈Rf(x+y)=f(x) \cdot f(y) \qquad \forall x,y \in \mathbb R Suppose that f(0)≠0f(0) \neq 0 and f(0)f(0) exists and it is finite (f(0)≠∞)(f(0) \neq \infty). Prove that ff has derivative in each point x∈R.x \in \mathbb R.
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