Let f:R → R satisfy f(x+y) = f(x) + f(y) for real numbers x and y

[MIT2014]

Homework Statement

Let f:R\rightarrowR satisfy f(x+y) = f(x) + f(y) for real numbers x and y. If we let f be continuous, show that \exists a real number b such that f(x) = bx.

Homework Equations

n/a

The Attempt at a Solution

Nooooo clue!

 

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Hint
Show that
f(x) = xf(1) when x is rational.

If f is continuous at c in R then f is continuous on R thus
f(x) = xf(1) for all x in R .

 

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In which case b= f(1). :-)

 

[MIT2014]

hmmm.
so f(x) = f(x-1) + f(1) = 2f(1) + f(x-2) = ... = f(x-x) + xf(1) = xf(x)
but that only works when x is an integer. How would you do it for when x is rational?

 

[╔(σ_σ)╝]

x = m/n.

nx =m

f(nx) = f(x)+ ...f(x) = nf(x)

f(m) = mf(1).
Thus,

nf(x) = mf(1).

f(x) = (m/n)f(1) = xf(1).

:-)