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A linear map between two topological vector spaces, such as normed spaces for example, is continuous (everywhere) if and only if there exists a point at which it is continuous, in which case it is even uniformly continuous. Consequently, every linear map is either continuous everywhere or else continuous nowhere.
(a different Weierstrass Function which is also continuous and nowhere differentiable) Nowhere differentiable continuous function proof of existence using Banach's contraction principle. Nowhere monotonic continuous function proof of existence using the Baire category theorem. Johan Thim. "Continuous Nowhere Differentiable Functions".
Image:BlankMap-World.png – World map, Robinson projection centered on the meridian circa 11°15' to east from the Greenwich Prime Meridian. Microstates and island nations are generally represented by single or few pixels approximate to the capital; all territories indicated in the UN listing of territories and regions are exhibited.
A continuous map from a closed ball of Euclidean space to its boundary cannot be the identity on the boundary. Similarly, the Borsuk–Ulam theorem says that a continuous map from the n -dimensional sphere to R n has a pair of antipodal points that are mapped to the same point.
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The isomorphisms are the isometries, i.e. metric maps which are injective, surjective, and distance-preserving. As an example, the inclusion of the rational numbers into the real numbers is a monomorphism and an epimorphism, but it is clearly not an isomorphism; this example shows that Met is not a balanced category .
A world map is a map of most or all of the surface of Earth. World maps, because of their scale, must deal with the problem of projection. Maps rendered in two dimensions by necessity distort the display of the three-dimensional surface of the Earth. While this is true of any map, these distortions reach extremes in a world map.
By Darboux's theorem, the derivative of any differentiable function is a Darboux function. In particular, the derivative of the function (/) is a Darboux function even though it is not continuous at one point. An example of a Darboux function that is nowhere continuous is the Conway base 13 function.