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piecewise continuous functions

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1: 6.16 Mathematical Applications
It occurs with Fourier-series expansions of all piecewise continuous functions. … …
2: 1.4 Calculus of One Variable
For an example, see Figure 1.4.1
See accompanying text
Figure 1.4.1: Piecewise continuous function on [ a , b ) . Magnify
If ϕ ( x ) is continuous or piecewise continuous, then …
3: 3.7 Ordinary Differential Equations
Let ( a , b ) be a finite or infinite interval and q ( x ) be a real-valued continuous (or piecewise continuous) function on the closure of ( a , b ) . …
4: 1.5 Calculus of Two or More Variables
A function f ( x , y ) is piecewise continuous on I 1 × I 2 , where I 1 and I 2 are intervals, if it is piecewise continuous in x for each y I 2 and piecewise continuous in y for each x I 1 . …
5: 1.8 Fourier Series
If a n and b n are the Fourier coefficients of a piecewise continuous function f ( x ) on [ 0 , 2 π ] , then …
6: 2.3 Integrals of a Real Variable
In addition to (2.3.7) assume that f ( t ) and q ( t ) are piecewise continuous1.4(ii)) on ( 0 , ) , and …
7: 1.14 Integral Transforms
If f ( t ) is continuous and f ( t ) is piecewise continuous on [ 0 , ) , then … If f ( t ) is piecewise continuous, then … Also assume that f ( n ) ( t ) is piecewise continuous on [ 0 , ) . … If f ( t ) and g ( t ) are piecewise continuous, then … If f ( t ) is piecewise continuous on [ 0 , ) and the integral (1.14.47) converges, then …
8: 18.2 General Orthogonal Polynomials
Here w ( x ) is continuous or piecewise continuous or integrable such that … This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials (§18.20(i)). … The measure is not necessarily absolutely continuous (i. … Nevai (1979, p.39) defined the class 𝒮 of orthogonality measures with support inside [ 1 , 1 ] such that the absolutely continuous part w ( x ) d x has w in the Szegő class 𝒢 . …
Monotonic Weight Functions
9: 10.43 Integrals
§10.43(i) Indefinite Integrals
§10.43(iii) Fractional Integrals
The Kontorovich–Lebedev transform of a function g ( x ) is defined as …
  • (b)

    g ( x ) is piecewise continuous and of bounded variation on every compact interval in ( 0 , ) , and each of the following integrals

  • 10: 18.18 Sums
    §18.18(i) Series Expansions of Arbitrary Functions
    Alternatively, assume f ( x ) is real and continuous and f ( x ) is piecewise continuous on ( 1 , 1 ) . … Assume f ( x ) is real and continuous and f ( x ) is piecewise continuous on ( 0 , ) . … Assume f ( x ) is real and continuous and f ( x ) is piecewise continuous on ( , ) . …
    Expansion of L 2 functions