Locally constant function
In mathematics , a function from a topological space into a set is called locally constant if for each there is a neighborhood of on which is constant .
properties
- Every constant function is also locally constant.
- Each locally constant function of in an arbitrary amount is constant, since contiguous and not by at least two disjoint open sets to overlap is.
- Every locally constant holomorphic function from an open set into the complex numbers is constant if a domain is, that is , if it is connected.
- In general, every locally constant function is constant on every connected component ; the converse also applies to locally connected spaces .
- A mapping from a topological space into a discrete space is continuous if and only if it is locally constant.
- Every mapping from a discrete space to any topological space is locally constant.
- The set of locally constant functions in a space naturally form a sheaf of commutative rings .
Examples
- The function defined by for and for is locally constant. (This implies that it is irrational , since such and are open sets that cover.)
- The function , defined by for and for , is also locally constant.
- The sign function is not locally constant.
- Step functions are not local, but piecewise constant