In the mathematical field of analysis, a well-known theorem describes the set of discontinuities of a monotone real-valued function of a real variable; all discontinuities of such a (monotone) function are necessarily jump discontinuities and there are at most countably many of them.
Usually, this theorem appears in literature without a name. It is called Froda's theorem in some recent works; in his 1929 dissertation, Alexandru Froda stated that the result was previously well-known and had provided his own elementary proof for the sake of convenience. Prior work on discontinuities had already been discussed in the 1875 memoir of the French mathematician Jean Gaston Darboux.
Definitions
Denote the limit from the left by
and denote the limit from the right by
If and exist and are finite then the difference is called the jump of at
Consider a real-valued function of real variable defined in a neighborhood of a point If is discontinuous at the point then the discontinuity will be a removable discontinuity, or an essential discontinuity, or a jump discontinuity (also called a discontinuity of the first kind). If the function is continuous at then the jump at is zero. Moreover, if is not continuous at the jump can be zero at if
Precise statement
Let be a real-valued monotone function defined on an interval Then the set of discontinuities of the first kind is at most countable.
One can prove that all points of discontinuity of a monotone real-valued function defined on an interval are jump discontinuities and hence, by our definition, of the first kind. With this remark the theorem takes the stronger form:
Let be a monotone function defined on an interval Then the set of discontinuities is at most countable.
Proofs
This proof starts by proving the special case where the function's domain is a closed and bounded interval The proof of the general case follows from this special case.
Proof when the domain is closed and bounded
Two proofs of this special case are given.
Proof 1
Let be an interval and let be a non-decreasing function (such as an increasing function). Then for any
Let and let be points inside at which the jump of is greater or equal to :
For any so that
Consequently,
and hence
Since we have that the number of points at which the jump is greater than is finite (possibly even zero).
Define the following sets:
Each set is finite or the empty set. The union
contains all points at which the jump is positive and hence contains all points of discontinuity. Since every is at most countable, their union is also at most countable.
If is non-increasing (or decreasing) then the proof is similar. This completes the proof of the special case where the function's domain is a closed and bounded interval.
Proof 2
So let is a monotone function and let denote the set of all points in the domain of at which is discontinuous (which is necessarily a jump discontinuity).
Because has a jump discontinuity at so there exists some rational number that lies strictly in between (specifically, if then pick so that while if then pick so that holds).
It will now be shown that if are distinct, say with then
If then implies so that
If on the other hand then implies so that
Either way,
Thus every is associated with a unique rational number (said differently, the map defined by is injective). Since is countable, the same must be true of
Proof of general case
Suppose that the domain of (a monotone real-valued function) is equal to a union of countably many closed and bounded intervals; say its domain is (no requirements are placed on these closed and bounded intervals). It follows from the special case proved above that for every index the restriction of to the interval has at most countably many discontinuities; denote this (countable) set of discontinuities by If has a discontinuity at a point in its domain then either is equal to an endpoint of one of these intervals (that is, ) or else there exists some index such that in which case must be a point of discontinuity for (that is, ). Thus the set of all points of at which is discontinuous is a subset of which is a countable set (because it is a union of countably many countable sets) so that its subset must also be countable (because every subset of a countable set is countable).