Conversely, suppose " f " ( " x " ) is piecewise continuous on the positive real numbers, taking a value halfway between the limit values at any jump discontinuities, and suppose the integral
22.
Given a confidence envelope for F, it can be shown that the CDF within the envelope that minimizes the variance begins on the lower envelope, has a jump discontinuity to the upper envelope, and then continues along the upper envelope.
23.
However, from your description which is not perfectly clear; what is the meaning of " negated " here ? there appears to be a jump discontinuity there .-- Lambiam 15 : 46, 23 July 2007 ( UTC)
24.
(The numerical value of } } is . ) Here runs over the nontrivial zeros of the zeta function, and is the same as, except that at its jump discontinuities ( the prime powers ) it takes the value halfway between the values to the left and the right:
25.
I could believe that a function with a jump discontinuity could be represented by a infinite series of some basis functions ( e . g ., the Dirac delta ), but that approximating a square wave with a Fourier periodic series would result in an extra spike jump discontinuity was counterintuitive.
26.
I could believe that a function with a jump discontinuity could be represented by a infinite series of some basis functions ( e . g ., the Dirac delta ), but that approximating a square wave with a Fourier periodic series would result in an extra spike jump discontinuity was counterintuitive.
27.
Often phase unwrapping is employed along either or both the time axis, ?, and frequency axis, ?, to suppress any jump discontinuity of the phase result of the STFT . The time index ? is normally considered to be " " slow " " time and usually not expressed in as high resolution as time " t ".
28.
However, " g " may have jump discontinuities, or may have derivative zero " almost " everywhere while still being continuous and increasing ( for example, " g " could be the Cantor function ), in either of which cases the Riemann Stieltjes integral is not captured by any expression involving derivatives of " g ".
29.
However, if " f " decays only polynomially quickly, then it may be necessary to use a further step of Clenshaw Curtis quadrature to obtain exponential accuracy of the remapped integral instead of the trapezoidal rule, depending on more details of the limiting properties of " f " : the problem is that, although " f " ( " L " cot? ) / sin 2 ( ? ) is indeed periodic with period ?, it is not necessarily smooth at the endpoints if all the derivatives do not vanish there [ e . g . the function " f " ( " x " ) = tanh ( " x " 3 ) / " x " 3 decays as 1 / " x " 3 but has a jump discontinuity in the slope of the remapped function at ? = 0 and ? ].
