L, there is a punctured neighbourhood C of p, such that implies This corresponds to the regular topological definition of continuity, applied to the subspace topology on and the restriction of f to Continuity The function is continuous at if and only if is defined at and If the function is continuous in if and only if, for every , is defined at and the limit of as tends to through is Every rational function , where and are polynomials, can be prolongated, in a unique way, to a function from to that is continuous in In particular, this is the case of polynomial functions, which take the value at if they are not constant. Also, if the tangent function is extended so that then is continuous in but cannot be prolongated further to a function that is continuous in Many elementary functions that are continuous in cannot be prolongated to functions that are continuous in This is the case, for example, of the exponential function and all trigonometric functions. For example, the sine function is continuous in but it cannot be made continuous at As seen above, the tangent function can be prolongated to a function that is continuous in but this function cannot be made continuous at Many discontinuous functions that become continuous when the codomain is extended to remain discontinuous if the codomain is extended to the affinely extended real number system This is the case of the function On the other hand, some functions that are continuous in and discontinuous at become continuous if the domain is extended to This is the case for the arctangent. As a projective range When the real projective line is considered in the context of the real projective plane, then the consequences of Desargues' theorem are implicit. In particular, the construction of the projective harmonic conjugate relation between points is part of the structure of the real projective line. For instance, given any pair of points, the point at infinity is the projective harmonic conjugate of their midpoint. As projectivities preserve the harmonic relation, they form the automorphisms of the real projective line. The projectivities are described algebraically as homographies, since the real numbers form a ring, according to the general construction of a projective line over a ring. Collectively they form the group PGL(2,R). The projectivities which are their own inverses are called involutions. A hyperbolic involution has two