Proof of continuity $f(x+y) = f(x) + f(y)$

Question

I am trying to prove that if I have $f:\mathbb{R} \to \mathbb{R}$ satisfying $\forall x,y\in\mathbb{R},f(x+y) = f(x) + f(y)$. Which is assumed continuous at $0$, that $f$ is continuous on $\mathbb{R}$

I am fairly sure it is, for that property seems to be a property of polynomials, and we know polynomials are continuous where defined(all reals)

Any ideas for rigor?

$|x-a| \lt \delta \implies |f(x+a)-f(x)-f(a)| \lt \epsilon$

No idea where to go from here, this is my first time doing $\epsilon-\delta$ stuff.

The only polynomials satisfying that property are those of the form $p(x) = ax$. — Santiago Canez, Apr 29 '14 at 22:53
Cauchy's Functional Equation, See: http://www.cofault.com/2010/01/hunt-for-addictive-monster.html — Alan, Apr 29 '14 at 23:12
Proving that an additive function $f$ is continuous if it is continuous at a single point. Pointers to a few more facts about such functions can be found here. — Martin Sleziak, Nov 02 '14 at 10:00
@MartinSleziak I don't think I realised at the time, but this is(the satisfied property) homomorphism isn't it. Thank you for the links — , Nov 02 '14 at 10:10
Yes, you are right. The condition is equivalent to saying that $f$ is a group homomorphism from $(\mathbb R,+)$ to $(\mathbb R,+)$. (I stressed the word group homomorphism, since $\mathbb R$ is not only a group, but also a ring, field, poset and many other things.) — Martin Sleziak, Nov 02 '14 at 10:23

Pedro · Accepted Answer · 2015-02-14T01:57:43.930

8

For any $x$, $$f(x+h)-f(x)=f(h)$$

Since $f$ is continuous at $x=0$, $f(h)\to 0$ as $h\to 0$.

edited Feb 14 '15 at 01:57

answered Apr 29 '14 at 22:54

Pedro

122,002

score 2 · Answer 2 · answered Apr 30 '14 at 01:20

Putting $x=y=0$, $f(0) = 0$.

Since $f$ is assumed to be continuous at $0$, $\lim_{x \to 0} f(x) = f(0) = 0$.

Finally, as Pedro Tamaroff wrote, since $f(x+h)-f(x) = f(h)$, $\lim_{h \to 0} f(x+h)-f(x) = \lim_{h \to 0} f(h) =0$, so $f$ is continuous at all $x$.

Proof of continuity $f(x+y) = f(x) + f(y)$

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