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A formal mathematical deﬁnition

Nick Pollock

This article attempts to express a deﬁnition of informal mathematics, the

continuity of a function, in the formal notation of ﬁrst-order logic. This is

useful when trying to prove the negation of a property whose deﬁnition is

complicated.

Informal deﬁnition of continuity

Let f : R → R be a function and δ,  ∈ R.

f is continuous at a ∈ R if, for any  > 0, there is a δ > 0 such

that |f(x) −f(a)| <  for all x satisfying |x −a| < δ.

The negation of this deﬁnition is

f is not continuous at a ∈ R if there is an  > 0 for which

there is no δ > 0 such that |f(x) − f(a)| <  for all x satisfying

|x − a| < δ.

or, equivalently,

f is not continuous at a ∈ R if there is an  > 0 such that

for any δ > 0 there is an x satisfying |x − a| < δ for which

|f(x) −f(a)| ≥ .

Formal deﬁnitions

Consider the theorem

If x is greater than 4, then x is greater than 3.

If no universe of discourse is given this statement is meaningless. What if

x is ‘a bowl of petunias’ for example? This is usually solved informally by

reducing the domain of x:

If x is an integer greater than 4, then x is greater than 3.

Now what about

If x is an integer greater than 4, then

√

x is greater than 2?

Most people are quite happy with this statement, but the implicit uni-

verse of discourse introduced in the ﬁrst part of the theorem, the integers,

does not always include

√

x, and the relation ‘greater than’ must be deﬁned

on more than the integers for the theorem to make sense. This problem is

particularly acute in theorems like the /N deﬁnition of convergence, where

some variables are real numbers and some are integers.

The next question is how to express this sort of statement in ﬁrst-order

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Motorola M500 Manual de usuario Pagina 22