Friday, October 2, 2015

real analysis - $f$ is linear and continuous at a point $implies f$ should be $f(x) =ax, $ for some $a in mathbb R$

Let $f$ be a real valued function defined on $\mathbb R$ such that $f(x+y)=f(x)+f(y)$.
Suppose there exists at least an element $x_0 \in \mathbb R$ such that $f$ is continuous at $x.$ Then prove that $f(x)=ax,\ \text{for some}\ x \in \mathbb R.$



Hints will be appreciated.

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