![]() ![]() Peano's solution does not set up a continuous one-to-one correspondence between the unit interval and the unit square, and indeed such a correspondence does not exist (see § Properties below). The problem Peano solved was whether such a mapping could be continuous i.e., a curve that fills a space. Peano was motivated by Georg Cantor's earlier counterintuitive result that the infinite number of points in a unit interval is the same cardinality as the infinite number of points in any finite-dimensional manifold, such as the unit square. His purpose was to construct a continuous mapping from the unit interval onto the unit square. In 1890, Peano discovered a continuous curve, now called the Peano curve, that passes through every point of the unit square. It is also possible to define curves without endpoints to be a continuous function on the real line (or on the open unit interval (0, 1)). Sometimes, the curve is identified with the image of the function (the set of all possible values of the function), instead of the function itself. In the most general form, the range of such a function may lie in an arbitrary topological space, but in the most commonly studied cases, the range will lie in a Euclidean space such as the 2-dimensional plane (a planar curve) or the 3-dimensional space ( space curve). A curve (with endpoints) is a continuous function whose domain is the unit interval.
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