F is continuous if and only if f is continuous at 0

[mathmari]

Hey!

Let $f : \mathbb{R}\rightarrow \mathbb{R}$ be a function with $ f(x + y) = f(x) + f(y)$ for all $x, y \in \mathbb{R}$.
I want to show that $f$ is continuous if and only if $f$ is continuous at $0$. I have done the following:

$\Rightarrow$ :
This direction is trivial. $f$ is continuous, so the function is continuous at each point, so also at $0$. $\Leftarrow$ :
Since $f$ is continuous in $0$, we have that $\displaystyle{\lim_{x\rightarrow 0}f(x)=f(0)}$.

For $x=y=0$ we have that $f(0+0)=f(0)=f(0)+f(0)=2f(0)\Rightarrow f(0)=2f(0)$, also $f(0)=0$.

So it holds that $\displaystyle{\lim_{x\rightarrow 0}f(x)=f(0)=f(0)}$.

Let $a\in \mathbb{R}$ be an arbitrary number. We want to show that $\displaystyle{\lim_{x\rightarrow a}f(x)=f(a)}$.

We set $y=x-a$. When $x\rightarrow a$, then $y\rightarrow 0$.

Solving for $x$ we get $x=y+a$.

So, we have the following:
\begin{equation*}\lim_{x\rightarrow a}f(x)=\lim_{y\rightarrow 0}f(y+a)=\lim_{y\rightarrow 0}[f(y)+f(a)]=\lim_{y\rightarrow 0}f(y)+\lim_{y\rightarrow 0}f(a)=\lim_{y\rightarrow 0}f(y)+f(a)=0+f(a)=f(a)\end{equation*}

Therefore, $f$ is continuous. Is everything correct? Also the part where I change the variable? (Wondering)

 

[Euge]

$\Leftarrow$ :
Since $f$ is continuous in $0$, we have that $\displaystyle{\lim_{x\rightarrow 0}f(x)=f(0)}$.

For $x=y=0$ we have that $f(0+0)=f(0)=f(0)+f(0)=2f(0)\Rightarrow f(0)=2f(0)$, also $f(0)=0$.

So it holds that $\displaystyle{\lim_{x\rightarrow 0}f(x)=f(0)=f(0)}$.

It appears you meant to say at the end $\lim_{x\to 0} f(x) = f(0) = \color{red}{0}$, which is correct.

Let $a\in \mathbb{R}$ be an arbitrary number. We want to show that $\displaystyle{\lim_{x\rightarrow a}f(x)=f(a)}$.

We set $y=x-a$. When $x\rightarrow a$, then $y\rightarrow 0$.

Solving for $x$ we get $x=y+a$.

So, we have the following:
\begin{equation*}\lim_{x\rightarrow a}f(x)=\lim_{y\rightarrow 0}f(y+a)=\lim_{y\rightarrow 0}[f(y)+f(a)]=\lim_{y\rightarrow 0}f(y)+\lim_{y\rightarrow 0}f(a)=\lim_{y\rightarrow 0}f(y)+f(a)=0+f(a)=f(a)\end{equation*}

Therefore, $f$ is continuous.

This is absolutely right.

 

[mathmari]

It appears you meant to say at the end $\lim_{x\to 0} f(x) = f(0) = \color{red}{0}$, which is correct.

Oh yes... (Blush)

This is absolutely right.

Great! Thank you! (Happy)