Analysis | Part IA, 2004

Let ff be defined on R\mathbb{R}, and assume that there exists at least one point x0Rx_{0} \in \mathbb{R} at which ff is continuous. Suppose also that, for every x,yR,fx, y \in \mathbb{R}, f satisfies the equation


Show that ff is continuous on R\mathbb{R}.

Show that there exists a constant cc such that f(x)=cxf(x)=c x for all xRx \in \mathbb{R}.

Suppose that gg is a continuous function defined on R\mathbb{R} and that, for every x,yRx, y \in \mathbb{R}, gg satisfies the equation

g(x+y)=g(x)g(y).g(x+y)=g(x) g(y) .

Show that if gg is not identically zero, then gg is everywhere positive. Find the general form of gg.

Typos? Please submit corrections to this page on GitHub.