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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: 1.8 Fourier Series
If a n and b n are the Fourier coefficients of a piecewise continuous function f ( x ) on [ 0 , 2 π ] , then …
5: 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 . …
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: 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

  • 9: 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 ( , ) . … For the modified Bessel function I ν ( z ) see §10.25(ii). …
    10: 18.2 General Orthogonal Polynomials
    A system (or set) of polynomials { p n ( x ) } , n = 0 , 1 , 2 , , is said to be orthogonal on ( a , b ) with respect to the weight function w ( x ) ( 0 ) if
    18.2.1 a b p n ( x ) p m ( x ) w ( x ) d x = 0 , n m .
    Here w ( x ) is continuous or piecewise continuous or integrable, and such that 0 < a b x 2 n w ( x ) d x < for all n . … More generally than (18.2.1)–(18.2.3), w ( x ) d x may be replaced in (18.2.1) by a positive measure d α ( x ) , where α ( x ) is a bounded nondecreasing function on the closure of ( a , b ) with an infinite number of points of increase, and such that 0 < a b x 2 n d α ( x ) < for all n . … This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials (§18.20(i)). …