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### A discontinuous function such that $f(x + y) = f(x) + f(y)$ [duplicate]

Is it possible to construct a function $f \colon \mathbb{R} \to \mathbb{R}$ such that $$f(x + y) = f(x) + f(y)$$ and $f$ is not continuous?
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### Function that satisfies $f(x+ y) = f(x) + f(y)$ but not $f(cx)=cf(x)$ [duplicate]

Is there a function from $\Bbb R^3 \to \Bbb R^3$ such that $$f(x + y) = f(x) + f(y)$$ but not $$f(cx) = cf(x)$$ for some scalar $c$? Is there one such function even in one dimension? I so, what is ...
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### Does a function that satisfies the equality $f(a+b) = f(a)f(b)$ have to be exponential? [duplicate]

I understand the other way around, where if a function is exponential then it will satisfy the equality $f(a+b)=f(a)f(b)$. But is every function that satisfies that equality always exponential?
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### 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 $\mathbb{R}$.
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### $f(a+b)=f(a)+f(b)$. Prove that $f(x)=Cx$, where $C=f(1)$ [duplicate]

A question from Introduction to Analysis by Arthur Mattuck: Suppose $f(x)$ is continuous for all $x$ and $f(a+b)=f(a)+f(b)$ for all $a$ and $b$. Prove that $f(x)=Cx$, where $C=f(1)$, as follows: (a)...
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### 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(...
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### If $f(x+y)=f(x)+f(y) ,\forall\;x,y\in\Bbb{R}$, then if $f$ is continuous at $0$, then it is continuous on $\Bbb{R}.$ [duplicate]

I know that this question has been asked here before but I want to use a different approach. Here is the question. A function $f:\Bbb{R}\to\Bbb{R}$ is such that \begin{align} f(x+y)=f(x)+f(y) ,\;\;\...
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### $f(x + y) = f(x) + f(y)$. Show that $f$ is continuous. [duplicate]

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ and $f(x + y) = f(x) + f(y)$. How can I show that $f$ is continuous, when $f$ is continuous at $f(0)$?
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### $f$ is linear and continuous at a point $\implies\ f$ should be $f(x) =ax,$ for some $a \in \mathbb R$ [duplicate]

Let $f$ be a real valued function defined on $\mathbb R$ such that $f(x+y)=f(x)+f(y)$. Suppose there exists at least an element $x_0 \in \mathbb R$ such that $f$ is continuous at $x.$ Then prove ...
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### If $f(x+y)=f(x)+f(y)$ in real numbers, does the function $f$ have to be continuous? [duplicate]

Given $f:\mathbb{R}\rightarrow\mathbb{R}$, and the function $f$ satisfies $f(x+y)=f(x)+f(y)$ for any $x,y\in S$. Can we say that this function $f$ must be continuous? I think it is false, but couldn'...
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### Functional Equation Proof [duplicate]

Question: If $f$ is a function such that $$f(x+y) = f(x)+f(y) \qquad f(xy) = f(x)f(y)$$ for all $x$, then prove that $f(x) = 0$ or $f(x) = x$ for all $x$. (The fact that every positive number is a ...
### Can we deduce that any additive map from V to W, over the field $R$ is actually a linear transformation? [duplicate]
I showed as an exercise that for an additive function $T: V\to W$, if $T(u+v)= T(u)+T(v)$, then if the field is the rationals, we get that the linearity implies scalar multiplication, i.e. \$T(qv)=qT(...