Some definitions of the concept of continuity

Euler (1784)

A continuous curve is one such that its nature can be expressed by a single function of x. If a curve is of such a nature that for its various parts ... different functions of x are required for its expression, ... , then we call such a curve discontinuous.

Bolzano (1817)

A function f(x) varies according to the law of continuity for all values of x inside or outside certain limits ... if [when] x is some such value, the difference f( x + ) - f(x) can be made smaller than any given quantity provided can be taken as small as we please.

Cauchy (1821)

The function f will be, between two assigned values of the variable x, a continuous function if for each value of x between these limits the [absolute] value of the difference f (x + ) - f (x) decreases indefinitely with .

Dirichlet (1837)

One thinks of a and b as two fixed values and of x as a variable quantity that can progressively take all values lying between a and b. Now if to every x there corresponds a single, finite y in such a way that, as x continuously passes through the interval from a to b, y = f(x) also gradually changes, then y is called a continuous function of x in this interval.

Heine (1872)

A function f(x) is continuous at the particular value x = X if for every quantity , no matter how small, there exists a positive number ₀ with the property that for no positive quantity which is smaller than ₀ does the absolute value of f (X h) - f (X) exceed . A function f (x) is continuous from x = a to x = b if for every single value x = X between a and b, including x = a and x = b, it is continuous.

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JOC September 2002