Paper 1, Section II, D

Analysis I | Part IA, 2019

Let g:RRg: \mathbb{R} \rightarrow \mathbb{R} be a function that is continuous at at least one point zRz \in \mathbb{R}. Suppose further that gg satisfies


for all x,yRx, y \in \mathbb{R}. Show that gg is continuous on R\mathbb{R}.

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

Suppose that h:RRh: \mathbb{R} \rightarrow \mathbb{R} is a continuous function defined on R\mathbb{R} and that hh satisfies the equation

h(x+y)=h(x)h(y)h(x+y)=h(x) h(y)

for all x,yRx, y \in \mathbb{R}. Show that hh is either identically zero or everywhere positive. What is the general form for hh ?

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