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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Continuity

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.

Definition

A function f is continuous at a number a if:

This definition implicitly requires that:

1. f(a) is defined (that is, a is in the domain of f)
2. lim_x→a⁡ f(x) exists
3. lim_x→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: lim_x→a⁡ f(x) does not exist

Case 3: lim_x→a⁡ f(x) ≠ f(a)