# Provided $f$ is continuous at $x_0$, and $f(x+y) = f(x) + f(y)$ prove $f$ is continuous everywhere. [duplicate]

My attempt...

By definition, whenever $|x- x_0| < \delta$ we have $|f(x) - f(x_0)| < \epsilon$. Observing that

\begin{align} |f(x) - f(y)| &= |f(x -x_0 + x_0) + f(y)| = |f(x-x_0) + f(x_0) - f(y)| \newline \newline &\leq \epsilon + |f(y) - f(x_0)| = \epsilon + |f(y-x_0)|... \end{align}

Here I need to choose a delta that can depends on $\epsilon$ and $y$ s.t. whenever $0<|x-y|< \delta$ then the above inequality is bounded by any $\epsilon$.

I'm also, in general, having trouble understanding this concept of continuity on an interval. I believe the structure of the definition is: for any $\epsilon> 0$ and any number $y$ in the interval, there exists a $\delta$ that depends on $\epsilon$ and $y$ such that for all $x$ in the interval and $|x - y | < \delta$ then $|f(x) - f(y)| < \epsilon$.

This definition makes me tempted to just choose y to be in the same delta neighborhood as $x$ in the given statement, but that constricts continuity to a small interval.

Edit: This question assumes no knowledge of Lebesgue measure theory.

• What is your question? Commented Feb 7, 2018 at 14:53
• Accidentally deleted it. Edited. Commented Feb 7, 2018 at 14:56
• More general here no continuity is required: math.stackexchange.com/questions/2067152/… Commented Feb 7, 2018 at 15:10
• @GuyFsone It is likely that the OP is unaware of measurability, so this question might not be considered a duplicate. Commented Feb 7, 2018 at 16:03
• – dxiv
Commented Feb 7, 2018 at 16:55

For any $\epsilon>0$ there is a $\delta>0$ such that if $|x-x_0| < \delta$ then $|f(x)-f(x_0)| < \epsilon$.

Since $f(x)-f(x_0) = f(x-x_0)$ we see that this can be written as for any $\epsilon>0$ there is a $\delta>0$ such that if $|h| < \delta$ then $|f(h)| < \epsilon$.

Now pick some other $x_1$ and note that since $f(x) -f(x_1) = f(x-x_1)$, then if $|x-x_1| < \delta$ we must have $|f(x)-f(x_1)| < \epsilon$.

Sketch/Scratch work:

• First, prove $$f(0)=0$$ since $$f(0+0)=f(0)+f(0)$$.

• Second, prove $$f(-y)=-f(y)$$ since $$f(y-y)=f(y)+f(-y)$$.

• Suppose that $$|x-y|<\delta$$. Then $$|f(x)-f(y)|=|f(x)-f(y)+f(x_0)-f(x_0)|=|f((x-y)+x_0)-f(x_0)|.$$

• Observe that since $$|x-y|<\delta$$, you have a version of the continuity statement for $$x_0$$ (replace $$x-y$$ by $$w$$ where $$|w|<\delta$$ if you don't see it).

More details:

Let $$\varepsilon>0$$, since $$f$$ is continuous at $$x_0$$, there exists a $$\delta$$ so that if $$|x-x_0|<\delta$$, then $$|f(x)-f(x_0)|<\varepsilon$$. Fix $$x_1$$ and let $$y$$ be such that $$|y-x_1|<\delta$$. If you can prove that $$|f(y)-f(x_1)|<\varepsilon$$, then you have proved continuity at $$x_1$$. Since the choice of $$x_1$$ is arbitrary, $$f$$ is continuous.

• I understand you're work except for your conclusion from the last absolute value, i.e., how does $|f(w + x_0) -f(x_0)|$ give a continuity statement for $x_0$? Commented Feb 7, 2018 at 16:29
• You have continuity at $x_0$ (you don't need to prove it). This statement allows you to transfer the continuity at $x_0$ to $x$. Commented Feb 7, 2018 at 16:40

Let $a\in\mathbb R.$ If the function $f(x)$ is continuous at $x_0,$ then the function $g(x):=f(x+a)=f(x)+f(a)$ is continuous at $x_0-a,$ and so is $f(x)=g(x)-f(a).$ Since $a$ is an arbitrary real number, so is $x_0-a;$ thus $f(x)$ is continuous everywhere.