Question

Suppose that f satisfies f(x+y)=f(x)+f(y), and that f is continuous at 0. Prove that f is continuous on R.

Answer 1

what is the definition of continuous for all \(a\in \mathbb R\) you have to show that
\[\lim_{x\to a}f(x)=f(a)\]
now
\[\lim_{x\to a}f(x)=\lim{h\to 0}f(a+h)=f(a)+\lim_{h\to 0}f(h)=f(a)+f(0)\]

the first equality by a change of variable,
the second by the condition that \(f(x+y)=f(x)+f(y)\)
and the third by the fact that \(f\) is continuous at \(x=0\)
all that is left for you show is that if \(f(x+y)=f(x)+f(y)\) then it must be true that \(f(0)=0\)

Answer 2

i will leave that up to you, but if you want a hint, start with
\[f(0)=f(0+0)\]

Answer 3

okay, i will try first, but if i don't know to start with, i will ask you again ya..