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In basic calculus an analysis we end up writing the words "continuous" and "differentiable" nearly as often as we use the term "function", yet, while there are plenty of convenient (and even fairly precise) shorthands for representing the latter, I'm not aware of a way to concisely represent the former.

Are there commonly used symbols or shorthands to represent "continuous" or "differentiable"?

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To say that a function $ f:X\to Y $ is continuous one writes $ f\in C( X,Y) $, which reads " f is in the set of continuous mappings from X to Y".

If a function $ f:X\to Y $ is continuously differentiable, one writes $ f\in C^{1} (X,Y). $

If it is $k$ times differentiable and that $k$-th derivative is continuous, one writes $ f \in C^{k} (X,Y).$

I don't think there is a common notation for a function which is differentiable, but whose derivative is not continuous. That doesn't seem to come up so often that we can't bear to write it in full in those cases though.

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The class $\mathcal{C}^0(D)$ or $\mathcal{C}(D)$ is the class of all continuous functions with domain $D$ (and codomain usually understood; if you want to specify the codomain, there are a number of possible notations, such as $\mathcal{C}^0(D,\mathbb{R})$ or $\mathcal{C}^0_{\mathbb{R}}(D)$). So one common shorthand for "$f$ is continuous" is $f\in\mathcal{C}^0$.

Differentiable is a bit harder; the class $\mathcal{C}^1$ requires that the derivative not only exists, but be continuous. So "$f'$ exists" is about as short-hand as you get.

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Whenever these notions are defined, people usually use $f\in C(A)$ to say that $f$ is continuous on $A$ (i.e. $f$ is continuous in any point $a\in A$) and $f\in C^k(A)$ to say that for any $a\in A$ there exists $f^{(k)}(a)$ and that $f^{(k)}$ is continuous on $A$.

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