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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);
Then f is a non-decreasing function on [a, b], which is continuous except for jump discontinuities at x n for n ≥ 1. In the case of finitely many jump discontinuities, f is a step function. The examples above are generalised step functions; they are very special cases of what are called jump functions or saltus-functions. [8] [9]
A jump discontinuity occurs when () (+), regardless of whether () is defined, and regardless of its value if it is defined. A removable discontinuity occurs when f ( c − ) = f ( c + ) {\displaystyle f(c^{-})=f(c^{+})} , also regardless of whether f ( c ) {\displaystyle f(c)} is defined, and regardless of its value if it is defined (but which ...
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); in an essential discontinuity, oscillation measures the failure of a limit to exist.
Jump to content. Main menu. ... 5 Can we define "x = a is a discontinuity of f(x) ... 7 Ambiguous wording about removable singularities and removable discontinuities.
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BV functions have only jump-type or removable discontinuities [ edit ] In the case of one variable, the assertion is clear: for each point x 0 {\displaystyle x_{0}} in the interval [ a , b ] ⊂ R {\displaystyle [a,b]\subset \mathbb {R} } of definition of the function u {\displaystyle u} , either one of the following two assertions is true