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piecewise continuous functions

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11: 1.17 Integral and Series Representations of the Dirac Delta
From the mathematical standpoint the left-hand side of (1.17.2) can be interpreted as a generalized integral in the sense that … for all functions ϕ ( x ) that are continuous when x ( , ) , and for each a , e n ( x a ) 2 ϕ ( x ) d x converges absolutely for all sufficiently large values of n . … More generally, assume ϕ ( x ) is piecewise continuous1.4(ii)) when x [ c , c ] for any finite positive real value of c , and for each a , e n ( x a ) 2 ϕ ( x ) d x converges absolutely for all sufficiently large values of n . … provided that ϕ ( x ) is continuous when x ( , ) , and for each a , e n ( x a ) 2 ϕ ( x ) d x converges absolutely for all sufficiently large values of n (as in the case of (1.17.6)). … provided that ϕ ( x ) is continuous and of period 2 π ; see §1.8(ii). …
12: 1.6 Vectors and Vector-Valued Functions
The path integral of a continuous function f ( x , y , z ) is …when f is continuous, and … The curve C is piecewise differentiable if 𝐜 is piecewise differentiable. … … The integral of a continuous function f ( x , y , z ) over a surface S is …
13: About Color Map
Surface visualizations in the DLMF represent functions of the form z = f ( x , y ) by the height z or the magnitude, | z | , for complex functions, over the x × y plane. … By painting the surfaces with a color that encodes the phase, ph f , both the magnitude and phase of complex valued functions can be displayed. …
Continuous Phase Mapping
For the continuous phase mapping, we map the phase continuously onto the hue, as both are periodic. … We therefore use a piecewise linear mapping as illustrated below, that takes phase 0 to red, π / 2 to yellow, π to cyan and 3 π / 2 to blue. …
14: 1.9 Calculus of a Complex Variable
Continuity
A function f ( z ) is continuous at a point z 0 if lim z z 0 f ( z ) = f ( z 0 ) . … If x ( t ) and y ( t ) are continuous and x ( t ) and y ( t ) are piecewise continuous, then z ( t ) defines a contour. … Let ( a , b ) be a finite or infinite interval, and f 0 ( t ) , f 1 ( t ) , be real or complex continuous functions, t ( a , b ) . …
15: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
for f ( x ) L 2 and piece-wise continuous, with convergence as discussed in §1.18(ii). … Eigenfunctions corresponding to the continuous spectrum are non- L 2 functions. … and completeness relation … More generally, for f C ( X ) , x X , see (1.4.24), …
§1.18(vii) Continuous Spectra: More General Cases