Linked Questions

2
votes
1answer
3k views

I want to show that $f(x)=x.f(1)$ where $f:R\to R$ is additive. [duplicate]

Possible Duplicate: Proving that an additive function $f$ is continuous if it is continuous at a single point Solution(s) to $f(x + y) = f(x) + f(y)$ (and miscellaneous questions…) I ...
3
votes
3answers
419 views

Proving that $f(x+y) = f(x) + f(y)$ and $f$ being continuous at one point implies that $f$ is continuous on ${\bf R}$ [duplicate]

Suppose $f(x+y)=f(x) + f(y)$ for all $x,y \in \mathbb R$ and $f$ is continuous at a point $a \in \mathbb R$. Prove that $f$ is continuous at every $b \in \mathbb R$. I know that in order to prove ...
0
votes
1answer
67 views

Conditions about continuous functions [duplicate]

Say we have $f(x+y) = f(x) + f(y) \quad \forall x,y \in \mathbb R$ and $f$ is continuous at one point at least. I wish to show there must be some $c$ such that $f(x)=cx$ for all $x$. Think I can do so ...
72
votes
1answer
12k views

Overview of basic facts about Cauchy functional equation

The Cauchy functional equation asks about functions $f \colon \mathbb R \to \mathbb R$ such that $$f(x+y)=f(x)+f(y).$$ It is a very well-known functional equation, which appears in various areas of ...
17
votes
2answers
3k views

If $f\colon \mathbb{R} \to \mathbb{R}$ is such that $f (x + y) = f (x) f (y)$ and continuous at $0$, then continuous everywhere

Prove that if $f\colon\mathbb{R}\to\mathbb{R}$ is such that $f(x+y)=f(x)f(y)$ for all $x,y$, and $f$ is continuous at $0$, then it is continuous everywhere. If there exists $c \in \mathbb{R}$ such ...
2
votes
2answers
601 views

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

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 $\mathbb{R}$ ...
3
votes
2answers
1k views

Alternatives to percentage for measuring relative difference?

If the value of something changes from $\;a\;$ to $\;b\;$, their relative difference can be expressed as a percentage: $$ \newcommand{\upto}{\mathop\nearrow} (D0) \;\;\; a \upto b \;=\; (b-a)/a \color{...
1
vote
3answers
208 views

If a function such that $f(x+y)=f(x)+f(y)$ is continuous at $0$, then it is continuous on $\mathbb R$ [duplicate]

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x+y)=f(x)+f(y)$. If $f$ is continuous at zero how can I prove that is continuous in $\mathbb{R}$.
1
vote
3answers
156 views

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(...
0
votes
1answer
170 views

Continuity of additive functions

I was checking this link Proving that an additive function $f$ is continuous if it is continuous at a single point and both Jspecter and Alex Becker's solutions seem to rely on the fact that $$\lim_{x\...
0
votes
1answer
40 views

Continuity limits

from a previous question asked here Proving that an additive function $f$ is continuous if it is continuous at a single point someone stated that the key step is that $\lim_{x \to c}f(x)=\lim_{x \to a}...