Continuity – (Part One).
Definition:
A function is continuous at a given point x = a if those three conditions below are met “simultaneously”:
(i) is defined. (i.e; a is in the domain of )
(ii) exists.
(iii)
NOTE:
- If any one of the three conditions is false, then is discontinuous at a, or it has a discontinuity at a.
Let’s now look at the different cases where may not be continuous at x = a.
(i) is defined but does not exist.
At a = 0, the function is not continuous despite is defined (Here, is equal to -1). This is because the two one-sided limits are not equal and as a consequence, the limit does not exist. This is called a jump discontinuity.
(ii) exists, but is not defined.
Assume a is the x-value where there is a hole in the graph. We can see that the limit from the right of a and the limit from the left of a are equal.…