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  2. Discontinuities of monotone functions - Wikipedia

    en.wikipedia.org/wiki/Discontinuities_of...

    Conversely, by a differentiation theorem of Lebesgue, the jump function f is uniquely determined by the properties: [14] (1) being non-decreasing and non-positive; (2) having given jump data at its points of discontinuity x n; (3) satisfying the boundary condition f (a) = 0; and (4) having zero derivative almost everywhere.

  3. Classification of discontinuities - Wikipedia

    en.wikipedia.org/wiki/Classification_of...

    The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function. The oscillation of a function at a point quantifies these discontinuities as follows: in a removable discontinuity, the distance that the value of the function is off by is the oscillation;

  4. Inflection point - Wikipedia

    en.wikipedia.org/wiki/Inflection_point

    For a function f, if its second derivative f″(x) exists at x 0 and x 0 is an inflection point for f, then f″(x 0) = 0, but this condition is not sufficient for having a point of inflection, even if derivatives of any order exist. In this case, one also needs the lowest-order (above the second) non-zero derivative to be of odd order (third ...

  5. Floor and ceiling functions - Wikipedia

    en.wikipedia.org/wiki/Floor_and_ceiling_functions

    At points of discontinuity, a Fourier series converges to a value that is the average of its limits on the left and the right, unlike the floor, ceiling and fractional part functions: for y fixed and x a multiple of y the Fourier series given converges to y/2, rather than to x mod y = 0. At points of continuity the series converges to the true ...

  6. Dirichlet–Jordan test - Wikipedia

    en.wikipedia.org/wiki/Dirichlet–Jordan_test

    In mathematics, the Dirichlet–Jordan test gives sufficient conditions for a complex-valued, periodic function to be equal to the sum of its Fourier series at a point of continuity. Moreover, the behavior of the Fourier series at points of discontinuity is determined as well (it is the midpoint of the values of the discontinuity).

  7. Critical point (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Critical_point_(mathematics)

    A critical point of such a curve, for the projection parallel to the y-axis (the map (x, y) → x), is a point of the curve where (,) = This means that the tangent of the curve is parallel to the y -axis, and that, at this point, g does not define an implicit function from x to y (see implicit function theorem ).

  8. Thomae's function - Wikipedia

    en.wikipedia.org/wiki/Thomae's_function

    is continuous at every irrational number, so its points of continuity are dense within the real numbers. Proof of continuity at irrational arguments Since f {\displaystyle f} is periodic with period 1 {\displaystyle 1} and 0 ∈ Q , {\displaystyle 0\in \mathbb {Q} ,} it suffices to check all irrational points in I = ( 0 , 1 ) . {\displaystyle I ...

  9. Macaulay's method - Wikipedia

    en.wikipedia.org/wiki/Macaulay's_method

    The starting point is the relation from Euler-Bernoulli beam theory = Where is the deflection and is the bending moment. This equation [7] is simpler than the fourth-order beam equation and can be integrated twice to find if the value of as a function of is known.