The symbols $\sim$ and $\propto$ are used to denote direct proportionality of two quantities.
Sometimes, a weaker statement is enough or desired, for example when one can only assume that an increase in one variable $x$ will lead to an increase in the dependent variable $y$.
Is there a common notation for this fact that $y$ is a monotonically increasing function of $f$?
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$\begingroup$I don't know of a single symbol like you speak of, but if you're just looking for a shorthand you can try something like $x_1 > x_2 \Rightarrow f(x_1) > f(x_2)$ and perhaps that will explain you're reasoning as, say, an intermediate step in a proof. Or maybe you want something like $\frac{\text{d}f}{\text{d}x} : \mathbb{R} \to \mathbb{R}^+$. My point is, you can write the property of monotonicity you wish to use explicitly in symbols.
$\endgroup$ $\begingroup$There's no standardized representation for that query, but if you want to denote the monotonicity of a function, let say $f(x):\mathbb{S}\rightarrow\mathbb{R}$ you can define a function provided $f(x)$ is differentiable $\forall\;x\in\mathbb{S}$,
$$g(x)=f'(x) \;\forall \; x\in \mathbb{S}$$
And accordingly, if the function is monotonically increasing then you can represent as,
$$g(x):\mathbb{S}\rightarrow\mathbb{R^{+}-{\{0\}}}$$
Else if it is decreasing then,
$$g(x):\mathbb{S}\rightarrow\mathbb{R^{-}-{\{0\}}}$$
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