Intermediate Value Theorem - Implications of Theorem in Real World

Implications of Theorem in Real World

The theorem implies that on any great circle around the world, the temperature, pressure, elevation, carbon dioxide concentration, or any other similar scalar quantity which varies continuously, there will always exist two antipodal points that share the same value for that variable.

Proof: Take f to be any continuous function on a circle. Draw a line through the center of the circle, intersecting it at two opposite points A and B. Let d be defined by the difference f(A) − f(B). If the line is rotated 180 degrees, the value −d will be obtained instead. Due to the intermediate value theorem there must be some intermediate rotation angle for which d = 0, and as a consequence f(A) = f(B) at this angle.

This is a special case of a more general result called the Borsuk–Ulam theorem.

Another generalization for which this holds is for any closed convex n (n>1) dimensional shape. Specifically, for any continuous function whose domain is the given shape, and any point inside the shape (not necessarily its center), there exist two antipodal points with respect to the given point whose functional value is the same. The proof is identical to the one given above.

The theorem also underpins the explanation of why rotating a wobbly table will bring it to stability (subject to certain easily-met constraints).

Read more about this topic:  Intermediate Value Theorem

Famous quotes containing the words implications of, implications, theorem, real and/or world:

    The power to guess the unseen from the seen, to trace the implications of things, to judge the whole piece by the pattern, the condition of feeling life in general so completely that you are well on your way to knowing any particular corner of it—this cluster of gifts may almost be said to constitute experience.
    Henry James (1843–1916)

    Philosophical questions are not by their nature insoluble. They are, indeed, radically different from scientific questions, because they concern the implications and other interrelations of ideas, not the order of physical events; their answers are interpretations instead of factual reports, and their function is to increase not our knowledge of nature, but our understanding of what we know.
    Susanne K. Langer (1895–1985)

    To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.
    Albert Camus (1913–1960)

    ... But all the feelings that evoke in us the joy or the misfortune of a real person are only produced in us through the intermediary of an image of that joy or that misfortune; the ingeniousness of the first novelist was in understanding that, in the apparatus of our emotions, since the image is the only essential element, the simplification which consists of purely and simply suppressing the factual characters is a definitive improvement.
    Marcel Proust (1871–1922)

    The world is a cow that is hard to milk,—life does not come so easy,—and oh, how thinly it is watered ere we get it!
    Henry David Thoreau (1817–1862)