Thread: Real Analysis Proof help dealing with continuity

If R->R satisfies f(x+y)=f(x)+f(y) for all x,y in R and 0 is in C(f), then f is continuous.


show me a great proof...

thanks in advance

Originally Posted by antoniowilliams46

If R->R satisfies f(x+y)=f(x)+f(y) for all x,y in R and 0 is in C(f), then f is continuous.


show me a great proof...

thanks in advance

A great one huh?

This is obvious since and thus . Thus, now there exists some such that . Make

thats all that i need???? nothing more???

Originally Posted by antoniowilliams46

thats all that i need???? nothing more???

You need to say more but that is the entire idea.

still need help on what to say in the proof

i still need a proof for this one

Do you understand Drexel proof? If you tell me where you get lost, I may be able to help you.

I dont know where to go after abs val of x-y is less than delta.

Let z=x-y. Then .

Another way to see it is: f is continuous at x iff f(x+h)=f(x)+o(1) as . So in this case, f(x+h) = f(x) + f(h) by linearity and f(h)=o(1) by continuity at 0.