Continuity of functions at a point
As we all know (even from my Boundaries Course), function
is continuous at a point
, When:
That is, when the left-hand limit of this function at this point is equal to the right-hand limit of the function at this point is equal to the value of the function at this point.
If any of the equality is not satisfied, the function is not continuous at a point , and the point is called the point of discontinuity .
In this naming, you can go a step further and DISTINGUISH the points of discontinuity. We do it like this:
Points of discontinuity of the first type
Point of discontinuity we call it a point of discontinuity of type I if the boundaries
are finite (i.e. they are simply numbers).
Additionally, if these limits are equal, then the point of discontinuity of type I is called removable .
Points of discontinuity of the II kind
Point of discontinuity we call it a type II discontinuity point if any of the boundaries is not finite (i.e. it simply equals infinity plus or minus).
Example 1
This function has a point
point of discontinuity (because the left-side boundary at this point is 0 and the right-side boundary is 1 ). This is a point of discontinuity of the first kind, because the left and right boundaries at this point are finite (0 and 1). This is not a removable point of discontinuity of the first type, because the boundaries are not equal.
Example 2
This function has a point
point of discontinuity (because the left and right limits at this point are not equal to the value of the function at this point). This is a point of discontinuity of the first kind, because the left and right boundaries are finite (and equal to 1). This is a removable point of discontinuity of the first type, because the left and right boundaries are equal.
Example 3
This function has a point point of discontinuity (because the left and right boundaries at this point are not equal). This is a point of type II discontinuity, because the left-hand boundary at this point is equal
.
