Proving that a function is constant from functional equation
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$ \int_0^x f(t)dt=\int_0^{ax}f(t)dt+ \int_0^{bx}f(t)dt$ implies $f$ constant
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If $a\operatorname{Re}(f(z))+b\operatorname{Im}(f(z))=c$ then f is constant
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If $\Re(f)$ is bounded then f is constant.
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Prove continuous functions $f$ and $g$ are equal
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How can I prove that this function is bounded using the given 2 hints?
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Functional Equation: Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$
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Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?
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Show $f$ is constant if $|f(x)-f(y)|\leq (x-y)^2$.
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Show that a continuous function has a fixed point
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Given $f$ is continuous and $f(x)=f(e^{t}x)$ for all $x\in\mathbb{R}$ and $t\ge0$, show that $f$ is constant function
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Functions $f(x)/g(x), g(x)/h(x),h(x)/f(x)$ are constant
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$f$ is constant if derivative equals zero
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Proof $f(x)\equiv 0$
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Functional equation $f\left(\frac{x+y}{2}\right)+f\left(\frac{2xy}{x+y}\right)=f(x)+f(y)$ implies $2f(\sqrt{xy})=f(x)+f(y)$
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Overview of basic facts about Cauchy functional equation
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Proving a function is constant, under certain conditions?
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Prove that every function that verifies $|f(x)-f(y)|\leq(x-y)^2$ for all $x,y$ is constant.