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REAL FUNCTIONS
Introduction
continuous
The notion of continuous function poses many problems for students, even in the case of a function f:A→R
where A is a subset of R (the set of all real numbers). In that case, it is defined using the standard ε- δ
definition. This can be given an expansive generalization to metric spaces, and even further a reconstructive
generalization to general topological spaces via the rule about inverse images of open sets (quite baffling to
some students).
Continuity is several related ideas
One can define what it means for a function
to be continuous at a point,
to be continuous on a set
to be continuous (that is, on its whole domain).
The relation among these ideas has subtleties; for example the function 1/x is continuous at every point at
which it is defined, but not continuous on R.
Metaphors for continuity are usually inaccurate
The various intuitions for continuity that students hear about are mostly incorrect.
Example
The well-known unbroken-graph intuition fails for functions such as
This function is not continuous at 0 but nowhere is there a "break" in the graph.
Continuity is hard to put in words
It is difficult to say precisely in words what continuity means. The ε- δ definition is logically complicated
with nested quantifiers and several variable. This makes it difficult to understand. and attempts to put the
ε- δ definition into words usually fail to be accurate. For example:
"A function is continuous if you can make the output change as little as you want by making the change in
the input small enough."
That paraphrase does not capture the subtlety that if a certain change in the input will work, then so must any
smaller change at the same place. See also map.
Citations: GraKnuPat8971, HocYou6113, KelRos92145, Kla78668,
References: This discussion draws from [], Chapter 2. See also [] and [].