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  2. Classification of discontinuities - Wikipedia

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

    The function in example 1, a removable discontinuity. Consider the piecewise function = {< = >. The point = is a removable discontinuity.For this kind of discontinuity: The one-sided limit from the negative direction: = and the one-sided limit from the positive direction: + = + at both exist, are finite, and are equal to = = +.

  3. Removable singularity - Wikipedia

    en.wikipedia.org/wiki/Removable_singularity

    A graph of a parabola with a removable singularity at x = 2. In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point.

  4. Singularity (mathematics) - Wikipedia

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

    The simplest example of singularities are curves that cross themselves. But there are other types of singularities, like cusps. For example, the equation y 2 − x 3 = 0 defines a curve that has a cusp at the origin x = y = 0. One could define the x-axis as a tangent at this point, but this definition can not be the same as the definition at ...

  5. Removable discontinuity - Wikipedia

    en.wikipedia.org/?title=Removable_discontinuity&...

    Download as PDF; Printable version; In other projects Appearance. move to sidebar hide. ... Redirect to: Classification of discontinuities#Removable discontinuity;

  6. Oscillation (mathematics) - Wikipedia

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

    For example, in the classification of discontinuities: in a removable discontinuity, the distance that the value of the function is off by is the oscillation; in a jump discontinuity, the size of the jump is the oscillation (assuming that the value at the point lies between these limits from the two sides);

  7. Cusp (singularity) - Wikipedia

    en.wikipedia.org/wiki/Cusp_(singularity)

    Consider a smooth real-valued function of two variables, say f (x, y) where x and y are real numbers.So f is a function from the plane to the line. The space of all such smooth functions is acted upon by the group of diffeomorphisms of the plane and the diffeomorphisms of the line, i.e. diffeomorphic changes of coordinate in both the source and the target.

  8. L'Hôpital's rule - Wikipedia

    en.wikipedia.org/wiki/L'Hôpital's_rule

    A simple but very useful consequence of L'Hopital's rule is that the derivative of a function cannot have a removable discontinuity. That is, suppose that f is continuous at a , and that f ′ ( x ) {\displaystyle f'(x)} exists for all x in some open interval containing a , except perhaps for x = a {\displaystyle x=a} .

  9. Residue (complex analysis) - Wikipedia

    en.wikipedia.org/wiki/Residue_(complex_analysis)

    The next example shows that, computing a residue by series expansion, a major role is played by the Lagrange inversion theorem. Let u ( z ) := ∑ k ≥ 1 u k z k {\displaystyle u(z):=\sum _{k\geq 1}u_{k}z^{k}} be an entire function , and let v ( z ) := ∑ k ≥ 1 v k z k {\displaystyle v(z):=\sum _{k\geq 1}v_{k}z^{k}} with positive radius of ...