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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; in a jump discontinuity, the size of the jump is the oscillation (assuming that the value at the point lies between these limits of the two sides); in an ...
A removable discontinuity occurs when () = (+), also regardless of whether () is defined, and regardless of its value if it is defined (but which does not match that of the two limits). A type II discontinuity occurs when either f ( c − ) {\displaystyle f(c^{-})} or f ( c + ) {\displaystyle f(c^{+})} does not exist (possibly both).
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.
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A function is continuous on an open interval if the interval is contained in the function's domain and the function is continuous at every interval point. A function that is continuous on the interval (, +) (the whole real line) is often called simply a continuous function; one also says that such a function is continuous everywhere.
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} .
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). [4] If the function is continuous at x {\displaystyle x} then the jump at x {\displaystyle x} is zero.
Great Picard's Theorem (meromorphic version): If M is a Riemann surface, w a point on M, P 1 (C) = C ∪ {∞} denotes the Riemann sphere and f : M\{w} → P 1 (C) is a holomorphic function with essential singularity at w, then on any open subset of M containing w, the function f(z) attains all but at most two points of P 1 (C) infinitely often.