Several years ago I did an overview video on continuity and continuous functions but did not go through in too much detail so I have done just that in this video. In this video I go over what it means for a function to be continuous as well as illustrate several cases when a function is discontinuous or not continuous.
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Mathematical definition of continuity corresponds closely with the meaning in everyday language
A continuous process is one that takes places gradually, without interruption or abrupt change.
A function f is continuous at a number a if:
This definition implicitly requires that:
- f(a) is defined (that is, a is in the domain of f)
- limx→a f(x) exists
- limx→a f(x) = f(a) exists
We say a function is discontinuous otherwise.
Discontinuous or not continuous cases
Case 1: f(a) is not defined
Case 2: limx→a f(x) does not exist
Case 3: limx→a f(x) ≠ f(a)