Formally, a homotopy between two continuous functions f and g
from a topological space X to a topological space Y is defined to be
a continuous function H:X\times [0,1] to Y from the product of the space X
with the unit interval [0, 1] to Y such that H(x,0)=f(x)
and H(x,1)=g(x) for all x members of X.
If we think of the second parameter of H as time then
H describes a continuous deformation of f into g:
at time 0 we have the function f and at time 1 we have the function g.
We can also think of the second parameter as a "slider control" that
allows us to smoothly transition from f to g as the slider moves from 0 to 1,
and vice versa.
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