Show that if f : R - R satisfies the equation f(x + y) = f(x) + f(y) for every x, y in R, and if f is continuous at (at least) one point, then there exists c such that f(x) = cx for all x in R.

Question

Show that if f : R -> R satisfies the equation f(x + y) = f(x) + f(y) for every x, y in R, and if f is continuous at (at least) one point, then there exists c such that f(x) = cx for all x in R.

amistre64 · Answer

it seems to be asking you to show that a linear function is a linear function ....

anonymous · Answer

There are a few hints, but I don't know where to start.

anonymous · Answer

Hint: Show that f is continuous everywhere; let c = f(1) and show that f(q) = cq for every
rational q; deduce that f(x) = cx for every real x.

anonymous · Answer

I really just need to start, and I'm sure I know how to continue once I started

amistre64 · Answer

i believe that there are thrms that would have been learned in the material to back up alot of the work.  But I have a class soon so I dont have the time needed to focus right now :(

anonymous · Answer

thanks anyway

anonymous · Answer

see first try to find derivative for function

amistre64 · Answer

It says to assume the function is linear to start with (or at least partly linear); and then show that the property f(cx) = c f(x) is a natural extension of it

anonymous · Answer

you will get derivative as some constant

anonymous · Answer

hence function is of linear form

anonymous · Answer

|dw:1379682029981:dw|