Continuity
A linear transformation between topological vector spaces, for example normed spaces, may be continuous. If its domain and codomain are the same, it will then be a continuous linear operator. A linear operator on a normed linear space is continuous if and only if it is bounded, for example, when the domain is finite-dimensional. An infinite-dimensional domain may have discontinuous linear operators.
An example of an unbounded, hence discontinuous, linear transformation is differentiation on the space of smooth functions equipped with the supremum norm (a function with small values can have a derivative with large values, while the derivative of 0 is 0). For a specific example, sin(nx)/n converges to 0, but its derivative cos(nx) does not, so differentiation is not continuous at 0 (and by a variation of this argument, it is not continuous anywhere).
Read more about this topic: Linear Map
Famous quotes containing the word continuity:
“Continuous eloquence wearies.... Grandeur must be abandoned to be appreciated. Continuity in everything is unpleasant. Cold is agreeable, that we may get warm.”
—Blaise Pascal (1623–1662)
“There is never a beginning, there is never an end, to the inexplicable continuity of this web of God, but always circular power returning into itself.”
—Ralph Waldo Emerson (1803–1882)
“If you associate enough with older people who do enjoy their lives, who are not stored away in any golden ghettos, you will gain a sense of continuity and of the possibility for a full life.”
—Margaret Mead (1901–1978)