The confusion about discontinuity
Whilst reading Mícheál Ó Searcóid’s book Metric Spaces, I found out about a nuance in the definition of continuity that I was not previously aware of, and something which may be taught incorrectly at high schools. M. Searcóid states that a function such as tan(x) is continuous (read the page here). The definition of continuity at a point is based on the fact that the function has a value at that point (if a function is continuous at x = a, then f(a) has a value in the expression |f(x)-f(a)|<ϵ). However, following M. Searcóid’s line of thought about continuous functions, it does not make sense to consider points at which a function is not defined. If we were asked to prove that the function is ‘discontinuous’ at a point, we would need to show that the condition for continuity at that point is false. And the negation* of the condition for continuity would not make sense at a point where the function value is not defined.…
The domain of a composite function
In this article I outline a systematic way of finding the domain of a composite function. A definition that can be used for this purpose follows:
The explanatory method which follows is to show how to use this definition in different examples.
Let and where .
Consider the same constraints as in Example 2.1, but with
Vaught, RL. 1995. Set theory: An introduction. 2nd edition. Boston: Birkhäuser.