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 inline such that

(B.1) Numbered Display Equation

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

(B.2) Numbered Display Equation

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) Numbered Display Equation

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

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