How to denote a set of functions

Question

Say there is an unknown function $h(x)$ $$\int_A^B h(x) = c$$ $A$, $B$ and $c$ are known. So $h(x)$ can have various forms on the range $[A,B]$. I want to know how to denote the set of functions for $h(x)$. I know the notation for a set is $\{...\}$.

So would it be: $\{h(x)|\int_A^B h(x) = c\}$? Or is there a different way to refer to a bunch of different possible functions?

I intend to narrow down this set by gradually introducing boundary restrictions/conditions. E.g. $h(x) \in \mathbb R$ and $h(x) = f(x)\cdot g(x)$ with $g(x)$ known.

Use ${h:[A,B]\longrightarrow{\Bbb R}\vert\int_A^B h(x)dx = 3}$. $h(x)$ is the value of $h$ in $x$. — Martín-Blas Pérez Pinilla, Feb 14 '14 at 07:21
The way you're writing the set is accurate. You might want to say $h\in C^0([A,B])$ (if your functions are continuous) or something similar, just to be clear. — Ian Coley, Feb 14 '14 at 07:55

score 1 · Accepted Answer · answered Apr 21 '14 at 15:14

1

Summarizing the comments: try

$\{h \in C([A,B])\vert\int_A^B h(x)\,dx = c\}$ or
$\{h \in L^1([A,B])\vert\int_A^B h(x)dx = c\}$

depending on what kind of functions you consider. Simply saying functions with integral equal to $3$ is usually ambiguous: there are different kinds of integrals.

Avoid writing $h(x)$ when you mean $h$.

answered Apr 21 '14 at 15:14

user127096

9,683

Is first variant posible to denote a set of continuous piecewise linear function? – Nick Oct 22 '16 at 09:41

How to denote a set of functions

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