Then it was assumed a continuous function can change sign only by vanishing; to-day we prove it.
The normal performance of these special functions is determined by their general and continuous function.
The power series represents a continuous function in its domain of convergence (the end-points may have to be excluded).
This illustrates the nature of the general and continuous function of these organs.
The integral is a continuous function of each of the end-values.
For example, see what has happened to the idea of continuous function.
If I really give my mind to the task, cannot I define a continuous function which is not differentiable?
If this hypothesis were not admitted there would no longer be any way of representing the probability by a continuous function.
By a “continuous function” of one variable we always mean a function which is continuous throughout an interval.
A function f : D -> E, where D and E are cpos, is continuous if it is monotonic and
f (lub Z) = lub f z | z in Z|f z | z in Z
for all directed sets Z in D. In other words, the image of the lub is the lub of any directed image.
All additive functions (functions which preserve all lubs) are continuous. A continuous function has a least fixed point if its domain has a least element, bottom (i.e. it is a cpo or a "pointed cpo" depending on your definition of a cpo). The least fixed point is
fix f = lub f^n bottom | n = 0..infinity