Riemannian Geometry
Continuity refers to the property of a function that intuitively means it doesn't have any jumps, breaks, or holes in its graph. In the context of mathematics, particularly in geometry and analysis, a function is continuous if small changes in the input result in small changes in the output. This concept is crucial when examining induced distance functions and smooth maps, as it ensures that these functions behave predictably under small perturbations, preserving the structure of the spaces involved.
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