Motive behind the proof in f(x+y)=f(x)+f(y) is continuous

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SUMMARY

The discussion centers on proving the continuity of a function f, defined by the property f(x+y) = f(x) + f(y) for all real x and y, under the condition that f is continuous at 0. The key insight is to demonstrate that if f is continuous at 0, then it must also be continuous at any point a by using the transformation h = x - a. This approach effectively translates the problem to a simpler form, allowing for a straightforward proof of continuity across the entire domain.

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albert1993
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Homework Statement


The problem is: Suppose f is a function with the property f(x+y)=f(x)+f(y) for x,y in the reals. suppose f is continuous at 0. show f is continuous everywhere.

I saw this post as an alternative solution that doesn't use epsilon-delta:
https://www.physicsforums.com/showpost.php?p=1987108&postcount=4

But I'm not sure how one would be motivated to set h=x-a...
I know it's a nice step and all that ultimately proves the statement, but what would be the main motivation for this?

NOTE: I understand this solution fully. I just want to know how one is motivated to think this way...
 
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The best answer is try to prove it by yourself. Forget the proof you just saw. Start by trying to prove limit f(x) as x->0 must be 0. Then try and prove it your own way.
 
Last edited:
We were given that f is continuous at x= 0 and want to show that it is continuous at x= a. So I would think the "translation" x-> x- a that maps a to 0 would be obvious.
 

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