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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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