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On Essential and Removable Discontinuities in Real and Complex Analysis

Differential Calculus

Recall the following definitions in (any) standard textbook on (undergraduate calculus):

ESSENTIAL DISCONTINUITY - Any discontinuity that is not removable. That is, a place where a graph is not connected and cannot be made connected simply by filling in a single point. Step discontinuities and vertical asymptotes are two types of essential discontinuities.

Formally, an essential discontinuity is a discontinuity at which the limit of the function does not exist.

Now compare that definition with:

REMOVABLE DISCONTINUITY - A hole in a graph. That is, a discontinuity that can be "repaired" by filling in a single point. In other words, a removable discontinuity is a point at which a graph is not connected but can be made connected by filling in a single point.

Formally, a removable discontinuity is one at which the limit of the function exists but does not equal the value of the function at that point; this may be because the function does not exist at that point.

These two definitions stem from the conceptual definition of a continuous function.  From Wikipedia:

In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous". An intuitive (though imprecise) idea of continuity is given by the common statement that a continuous function is a function whose graph can be drawn without lifting the chalk from the blackboard.

Continuity of functions is one of the core concepts of topology. The general case for continuity of functions between two metric spaces is considered in real analysis (and also in complex analysis).  In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity.  Other forms of continuity also exist.

As an example, consider the function h(t) which describes the height of a growing flower at time t. This function is continuous. In fact, there is a dictum of classical physics which states that in nature everything is continuous. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous. (However, if one assumes a discrete set as the domain of function M, for instance the set of points of time at 4:00 PM on business days, then M becomes continuous function, as every function whose domain is a discrete subset of reals is.)

Advanced Calculus

Now, we have the following theorem from advanced calculus:

Theorem L.  Differentiability implies continuity.

See the following analogous theorem from the Math Forum:

*** If a function is differentiable in an interval then it must be continuous in that interval. ***

Question: Why? The basic definition of the derivative is as a special limit. In evaluating the special limit, say, as x approaches a, we need never consider the value x = a. So a limit can exist in an interval even though we have a point of discontinuity at the point (a, f(a)). Thus I conclude that if a function is differentiable everywhere then it CAN be discontinuous at (many) points. It is clear to me that if a function is continuous everywhere it need not be differentiable everywhere. For example y = the absolute value of x.

Therefore, a function which is differentiable everywhere is also continuous everywhere.

Real Analysis versus Complex Analysis

In real analysis, if a function f(x) is differentiable, its derivative f'(x) may not even be continuous.

In contrast, in COMPLEX ANALYSIS, differentiability of f(z) in some region implies that f'(z) is also differentiable there, and hence f'(z) is infinitely differentiable.  Here again, however, differentiability of f'(z) does not imply continuity of f'(z).

Now, in defining differentiability of a complex-valued function f(z) defined on a set Z, the derivative

f'(z_0) = lim_{z --> z_0} (f(z) - f(z_0))/(z - (z_0))

is naturally taken with respect to values z in the set Z.  The limit will therefore have a different meaning depending on whether z_0 is an interior point or a boundary point of Z.  The best way to avoid this complication is simply to require that every differentiable function be defined on an open set.  It is further advantageous if all functions are defined on a connected set.  For, if a function is defined on a set consisting of a number of disjoint components, then the values of the function on the different components really have nothing to do with the other, and it is more reasonable to consider the function on each component separately.

A function f(z) is said to be analytic in a region Z if it is defined and differentiable at each point of Z.  A function f(z) is analytic on an arbitrary point set E if it is analytic in some region that contains E.  In particular, "f(z) is analytic at a point z_0" means that f(z) is defined and differentiable in some neighborhood of z_0.

In general, the terms analytic, regular, holomorphic, and monogenic are equivalent, although slight niceties of difference exists as used by some authors.

If f(z) is analytic on Z, then the real and imaginary parts of f(z) satisfy the Cauchy-Riemann equations there.  The argument is not reversible.  Even if the Cauchy-Riemann equations are satisfied, it could still happen that differentiability fails there.

Clearly, we need additional conditions for the analyticity of a function f(z).  In the following theorem a useful sufficient condition is given, although weaker conditions are known.  (We shall have much occasion to become involved with such delicate problems.)

Theorem R.  If the Cauchy-Riemann equations are satisfied in some neighborhood of a point (x_0, y_0), and if u_x, u_y, v_x, v_y are continuous in this neighborhood, then

f(z) = u(x, y) + i v(x, y)

is analytic at z_0 = x_0 + i y_0.

(Reference:  Classical Complex Analysis by Liang-shin Hahn and Bernard Epstein)
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