Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

B.1 CONTINUITY AND BOUNDED VARIATION

Definition: Function Function y = g(x) is a mapping from x to y.

This definition implies that for a given x, the mapping to y is unique. The inverse image x = g−1(y) may not be unique; if it is unique, then g(x) is one-to-one.

Definition: Continuous Function Function g(x) is continuous at xo if for every δ > 0, there exists such that

(B.1)

We can also write for a continuous function g(x) at xo that

(B.2)

where in this case ε is positive or negative: the limit must hold approaching xo from the right and the left.

Thus, we can simply write that g(x) is continuous at xo if

(B.3)

for finite g(xo). The definition in (B.1) applies to a specific point xo, meaning that a particular pair apply to that point. This can be emphasized by writing δ as a function of xo and ε: (not to be confused with the Dirac delta function defined later). For some other point ...