A.1. Non-decreasing functions

DEFINITION A.1 (Non-decreasing function).– A function is non-decreasing if

A.2. Infimum and supremum

DEFINITION A.2 (Infimum, Supremum).– Let . Then,

In particular, we have the following properties:

PROPOSITION A.1.– Let . Then:

1. 1)
2. 2)
3. 3)

PROPOSITION A.2 (Inf/Sup decomposition).– Let be two sets.

A.3. Continuity

DEFINITION A.3 (Left- and right-limits).– Let and . If they exist, then we denote the right-limit and the left-limit of f at x:

[A.1]

DEFINITION A.4 (Continuity).– The function f is said to be left-continuous at x if f (x–) exists and f (x–) = f (x), which is equivalent to requiring that for any there exists such that for any .

Symmetrically, the function f is said to be right-continuous at x if f(+–) exists and f(x+) = f (x).

The function f is continuous at x if it is both left- and right-continuous.

The function f is left-continuous (respectively right-continuous or continuous) if the property holds for any .

We call a discontinuity of f any value such that or .

DEFINITION A.5 (Piecewise continuity).– A function f is said to be piecewise continuous if it has a finite number of discontinuities at any finite interval.

Note that some authors such as those in [BRO 07, section 2.2.5.2] require that a piecewise continuous function has a finite number of discontinuities at any interval. With our definition, the ceiling function is a piecewise continuous function, whereas it has an infinite number of discontinuities (for each integer value, see Figure A.1(a). However, it has at most discontinuities in any interval of length ℓ. On the contrary, Figure A.1(b) shows a function that is not piecewise continuous: function g(x) is defined by g(x) = x_i for , with , and g(x) = 1 for x > 1 has an infinite number of segments in the interval [0, 1], and so it is not piecewise continuous.

DEFINITION A.6 (Right- and left-continuous closure).– Let be a piecewise continuous function. Then, its right-continuous extension f_r (respectively left-continuous extension f_l) is defined as

Since f is piecewise continuous, it allows left and right limits at any point x.

PROPOSITION A.3 (Stability of continuity by sum, min, max).– Let be two functions. For any , if f (x–) and g(x–) exist, and the same for the right-limit.

If f and g are left-continuous (respectively right-continuous, respectively continuous) at x, then so are f + g, f – g, .