If $f(x+y)=f(x)+f(y)$ in real numbers, does the function $f$ have to be continuous?
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Overview of basic facts about Cauchy functional equation
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If $f$ is continuous and $f(x+y) = f(x)+f(y)$, then $f(x) = cx$ for all $x \in \mathbb{R}$
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Why aren’t continuous functions defined the other way around?
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Let $f:[1,10]\to \Bbb{Q}$ be a continuous function and $f(1)=10,$then $f(10)=?$
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Is a function defined at a single point continuous?
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If $f$ continuous and $\lim_{x\to-\infty }f(x)=\lim_{x\to\infty }f(x)=+\infty $ then $f$ takes its minimum.
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If $\lim_{x\to\infty} [f(x+1)-f(x)] =l$ then $\lim_{ x\to\infty}f(x)/x =l$ ($f$ is continuous)
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Let $f:[0,1] \rightarrow \mathbb{R}$ be a continuous function. If $f(0)=f(1)$ then exists $x\in [0,1]$ s.t. $f(x) = f(x+\frac{1}{2})$
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Real Analysis Question: continuity of $f(x) = \sum_{q_n<x} 1/n^2$
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Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
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Is the convolution of two continuous functions continuous?
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How to show $f:Y\to X^{+}$ is continuous?
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Can a function with just one point in its domain be continuous?
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$f : S^1 \to\mathbb R$ is continuous then $f(x)=f(-x)$ for some $x\in S^1$
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I need to find all functions $f:\mathbb R \rightarrow \mathbb R$ which are continuous and satisfy $f(x+y)=f(x)+f(y)$
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The continuity of $f(x,y) = \frac {\sin (xy)} {\sqrt {x^2 + y^2}}$ at $(0,0)$
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Is this function uniformly continuous?