Provided $f$ is continuous at $x_0$, and $f(x+y) = f(x) + f(y)$ prove $f$ is continuous everywhere.
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Overview of basic facts about Cauchy functional equation
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I need to prove the continuity of $f(x)=\log x$ using a $\epsilon-\delta$ proof
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If $f$ is nonnegative and integrable then $F(x) = \int_{-\infty}^x f$ is continuous.
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Proving that an additive function $f$ is continuous if it is continuous at a single point
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Proof (epsilon delta) for the continuity of a function at a point
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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 $\Bbb{R}.$
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proof that $f'$ is continuous at $a$
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Let $F : X → X$ be continuous. Prove that the set $\{x ∈ X : F(x) = x\}$ of fixed points of F is closed in X
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If $f$ is continuous and with a limit at infinity then $f$ is uniformly continuous
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Show f is continuous on (0, ∞) iff f is continuous at 1 for a defined function.
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$f(x^2) = 2f(x)$ and $f(x)$ continuous
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Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
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Show that the function $f = \frac{xy}{x^2 + y^2}$ is continuous along every horizontal and every vertical line
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Suppose $f: \mathbb{R} \to \mathbb{R}$ is function such that $f^3$ is continuous on $\mathbb{R}$. Prove that $f$ is continuous on $\mathbb{R}$
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Prove that any polynomial function is continuous
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Functional equation $f(xy)=f(x)+f(y)$ and continuity
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non-continuous function satisfies $f(x+y)=f(x)+f(y)$