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1: 6.16 Mathematical Applications
It occurs with Fourier-series expansions of all piecewise continuous functions. … …
2: About Color Map
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. …
3: 1.4 Calculus of One Variable
If f ( x ) is continuous on an interval I save for a finite number of simple discontinuities, then f ( x ) is piecewise (or sectionally) continuous on I . For an example, see Figure 1.4.1Continuity, or piecewise continuity, of f ( x ) on [ a , b ] is sufficient for the limit to exist. … If ϕ ( x ) is continuous or piecewise continuous, then … If f ( x ) is continuous or piecewise continuous on [ a , b ] , then …
4: 1.6 Vectors and Vector-Valued Functions
The curve C is piecewise differentiable if c is piecewise differentiable. … … Sufficient conditions for this result to hold are that F 1 ( x , y ) and F 2 ( x , y ) are continuously differentiable on S , and C is piecewise differentiable. … Suppose S is a piecewise smooth surface which forms the complete boundary of a bounded closed point set V , and S is oriented by its normal being outwards from V . …
5: 1.8 Fourier Series
If f ( x ) is of period 2 π , and f ( m ) ( x ) is piecewise continuous, then … For f ( x ) piecewise continuous on [ a , b ] and real λ , … If a n and b n are the Fourier coefficients of a piecewise continuous function f ( x ) on [ 0 , 2 π ] , then …
6: 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 . … Sufficient conditions for the limit to exist are that f ( x , y ) is continuous, or piecewise continuous, on R . … Moreover, if a , b , c , d are finite or infinite constants and f ( x , y ) is piecewise continuous on the set ( a , b ) × ( c , d ) , then …
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, and such that 0 < a b x 2 n w ( x ) d x < for all n . …
9: Bibliography D
  • G. C. Donovan, J. S. Geronimo, and D. P. Hardin (1999) Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets. SIAM J. Math. Anal. 30 (5), pp. 1029–1056.
  • 10: Bibliography L
  • X. Li, X. Shi, and J. Zhang (1991) Generalized Riemann ζ -function regularization and Casimir energy for a piecewise uniform string. Phys. Rev. D 44 (2), pp. 560–562.