Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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"?

share|improve this question

3 Answers 3

up vote 5 down vote accepted

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.

share|improve this answer

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.

share|improve this answer

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 $A$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.