mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a function

f: \mathbb \to \mathbb

is symmetrically continuous at a point ''x'' if :

\lim_ f(x+h)-f(x-h) = 0.

The usual definition of continuity implies symmetric continuity, but the converse is not true. For example, the function

x^

is symmetrically continuous at

x=0

, but not continuous. Also, symmetric differentiability implies symmetric continuity, but the converse is not true just like usual continuity does not imply differentiability. The set of the symmetrically continuous functions, with the usual

scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector ...

can be easily shown to have the structure of a

vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

over

\mathbb

, similarly to the usually continuous functions, which form a

linear subspace In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping''); * linearity of a ''polynomial''. An example of a li ...

within it.

References

* Differential calculus Theory of continuous functions Types of functions {{mathanalysis-stub