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Math and science::Analysis::Tao::09. Continuous functions on R

Left and right limits and discontinuities

Let XR and let x0X. Let f:XR be a function.

If f(x0) and f(x0+) both exist and are both equal to f(x0), then f is continuous at x0.

x0 is an element of X by definition, so f(x0) must exist and be defined as f's range includes all of X.

x0 must be an adherent point of both X(,x0) and X(x0,), as these are requirements for the left and right limits to exist.

Discontinuities

Note that we require both ① limits to exist and ② be equal to each other and ③ to f(x0), and ④ that the statement is only an implication, not a bi-implication. Discontinuities can be found by looking at these properties. Four discontinuities that arise at (at least):

  1. Asymptotic discontinuity and oscillatory discontinuity
  2. Jump discontinuity
  3. Removable discontinuity
  4. Domain break discontinuity, with function continuity maintained (I made up this name). Not 100% if it exists.

These are discussed further on the back side.


Example

1. A left or right limit doesn't exist

Two examples of discontinuites that arise from a limit not existing:

Asymptotic discontinuity
One or both limits do not exist as, informally, f(x) tends to infinity as xx0. Example: f:R{0},f(x):=1x has a discontinuity at 0.
Oscillatory discontinuity
A function can remain bounded but still not have a left or right limit near a point. For example: f:RR where f(x) is 1 if xQ or 0 otherwise.

2. Jump discontinuity

Where both limits exist but are not equal, we have a jump discontinuity. At least one must not be equal f(x0). For example, f:RR defined by:

f(x):={0if x01if x>0

3. Removable discontinuity

Let an right limits can exist and be equal to f(x0), yet f still might not be continuous. For example, f:RR defined by:

f(x):={1if x=00if x0

Then both left and right limits exist and we have f(x0)=f(x0+)f(x0). This is called a removable discontinuity.

4. (Not sure what called?) Domain break discontinuity

If X=[0,1][3,4] and f:XR is given by f(x):=2x (or some other function continuous on R), then surprisingly, according to how Tao defines continuity, f is continuous at all endpoints x=0, x=1, x=3 and x=4. This is so as for any ε>0 we can find a δ>0 such that |f(i)f(a)|<ε for all i{xX(aδ,a+δ)}.

Considering this example, continuity might best be thought of as the question of whether for all sequences (an)n=0 that converge to x0 it can be said that (f(an))n=0 converges to f(x0). In this view, we are forced to ignore the parts of the domain that are not connected to x0 (what is the techical term for not connected?).


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