Continuity in mathematics is the first derivative equal to zero or the Boundary condition.
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This normally refers to continuity as a property of certain functions (mappings). They are called continuous if the output depends in a certain way on the input in that a small alteration in the input only leads to a small alteration in the output. Continuous functions can intuitively be drawn with a pencil without ever stopping and beginning again. More formally, a map between two topological spaces is continuous if the preimage of every open set in the codomain is an open set in the domain.
MATHEMATICS
The "grandfather of mathematics" is often considered to be the ancient Greek mathematician Euclid. Euclid is known for his work "Elements," a mathematical and geometric treatise that has had a profound influence on the development of mathematics. His systematic approach to geometry and his emphasis on logical reasoning laid the foundation for much of modern mathematics.
It depends on who "he" is (or was).
There are basically SEVERAL continuity equations, one for each conserved quantity. The equations themselves are simply statements that matter (in the example of conservation of mass) will not appear out of nothing, or suddenly teleport to a far-away place.