By Nicolaas Du Plessis

ISBN-10: 0050020544

ISBN-13: 9780050020548

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**Example text**

That is, f is discontinuous at a point b not in its domain E if every interval (b − δ, b + δ) contains points of E. The function f defined by f (x) = 1/x has this sort of a discontinuity at 0. While 0 isn’t in the domain√ of f , every interval (−δ, δ) contains points that are. While we consider 1/ x to be discontinuous at 0, we don’t consider it to be discontinuous at −1 because an√ interval (−1 − δ, −1 + δ) need not contain any points in the domain of 1/ x. According to our definitions, a function is either continuous or discontinuous at any given point in the domain.

Mathematical operations don’t just happen all by themselves; something must be operated on. And the operation must result in something, even if that something is a concept we associate with nothing, such as zero or the empty set. So we naturally associate mathematical operations with two sets, one for the input and one for the output. ) The operation is called a function if the result of performing it depends only on the designated input, with a special emphasis on the word designated. For example, if the set for the input is a set of coins and the operation to be performed on a coin consists of determining its denomination, that’s a function.

Indeed, to prove a given function f is discontinuous at a given point a in its domain, we usually must produce such an ε. But we also consider f to be discontinuous where points are missing from its domain. That is, f is discontinuous at a point b not in its domain E if every interval (b − δ, b + δ) contains points of E. The function f defined by f (x) = 1/x has this sort of a discontinuity at 0. While 0 isn’t in the domain√ of f , every interval (−δ, δ) contains points that are. While we consider 1/ x to be discontinuous at 0, we don’t consider it to be discontinuous at −1 because an√ interval (−1 − δ, −1 + δ) need not contain any points in the domain of 1/ x.

### An introduction to potential theory by Nicolaas Du Plessis

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