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Continuous data is data which can take any numerical value within certain restrictions. It is data which is not discrete.Examples of continuous quantities are height and time. Examples of discrete quantities are shoe size, number of eggs.
A function of x is continuous if a small change in x only causes a small change in the function's value, not a sudden jump.

f(x) is continuous at x=a if:

∀δ > 0, ∃ε( δ) : |x−a| < ε( δ) \implies |f(x)−f(a)| < δ

i.e. we can make f(x) as close as we like to f(a) by making x close to a.

A map between topological spaces is continuous if the inverse image of an open set is open (or equivalently if the inverse image of a closed set is closed). Continuous maps preserve connectedness and compactness.
Of a map between metric spaces: for f : X → Y with (X,d) and (Y, ρ) metric spaces, given x ∈ X , ∀ε > 0 ,     ∃δ > 0 :     ∀y ∈ { y | d(x,y) \lt δ} ,     ρ(f(x),f(y)) \lt ε means that f is continuous at x ; if applied throughout a given region this agrees with the definition above. Compare uniformly continuous.
In the context of a normed vector space, a map is continuous if whenever x_n → x , f(x_n) → f(x) .