Let $f : R\rightarrow R$ be a function with the property $f(x + y) = f(x) + f(y)$ for all $x,y\in R$.
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Overview of basic facts about Cauchy functional equation
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If $f: [0,1]\rightarrow \mathbb{R}$ is continuous function positive
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Proof that continuous function on $ [0, 2]$ has $f(x) = f(x-1)$ for some $ x$
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Let $A$ be the set of all functions $f:\mathbb{R}\to\mathbb{R}$ that satisfy the following two properties.
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Finding functions $f: \Bbb R_*^+ \to \Bbb R_*^+$ with certain properties
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Find all functions $ f : \Bbb R \to \Bbb R$ which satisfy $f(x)f(y) = f(x + y) + xy$ for all $x, y ∈ \Bbb R$.
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non-continuous function satisfies $f(x+y)=f(x)+f(y)$
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Solve for all possible functions f: $|f(x)-f(y)|=2|x-y|$.
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Suppose f is a function such that f'(x) = 1/x for all x > 0. Prove that if f(1) = 0, then f(xy) = f(x) + f(y) for all x,y > 0
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Prove that every function that verifies $|f(x)-f(y)|\leq(x-y)^2$ for all $x,y$ is constant.
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Functional equation $f(xy)=f(x)+f(y)$ and continuity
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$f(f(x)f(y))+f(x+y)=f(xy)$
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Prove: $f: \mathbb{R} \rightarrow \mathbb{R}$ st for every $x \in \mathbb{R}$ there exists $n$ st $f^{(n)}(x) = 0$, f is a polynomial.
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$f(x)f(1/x)=f(x)+f(1/x)$
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If $f(x + y) \leq yf(x) + f(f(x))$ for all real numbers $x$ and $y$, then prove that $f(x) = 0$ for all $x ≤ 0$.
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Find $f:f(xf(x)+f(y))=f(x)^2+y$
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$f \text{ convex }\Rightarrow \forall (x_0,f(x_0))\in \Gamma(f) \exists a : \forall x\in \mathbb{R}, f(x) \ge a(x-x_0)+f(x_0)$