About the Project
NIST

continuous function

AdvancedHelp

(0.002 seconds)

1—10 of 78 matching pages

1: 1.4 Calculus of One Variable
§1.4(ii) Continuity
See accompanying text
Figure 1.4.1: Piecewise continuous function on [ a , b ) . Magnify
If f ( x ) C n + 1 [ a , b ] , then …
2: 4.12 Generalized Logarithms and Exponentials
For C generalized logarithms, see Walker (1991). …
3: 2.8 Differential Equations with a Parameter
in which ξ ranges over a bounded or unbounded interval or domain Δ , and ψ ( ξ ) is C or analytic on Δ . … Again, u > 0 and ψ ( ξ ) is C on ( α 1 , α 2 ) . Corresponding to each positive integer n there are solutions W n , j ( u , ξ ) , j = 1 , 2 , that are C on ( α 1 , α 2 ) , and as u Also, ψ ( ξ ) is C on ( α 1 , α 2 ) , and u > 0 . … In the former, corresponding to any positive integer n there are solutions W n , j ( u , ξ ) , j = 1 , 2 , that are C on ( 0 , α 2 ) , and as u
4: 3.5 Quadrature
where h = b - a , f C 2 [ a , b ] , and a < ξ < b . … If in addition f is periodic, f C k ( ) , and the integral is taken over a period, then … Let h = 1 2 ( b - a ) and f C 4 [ a , b ] . … If f C 2 m + 2 [ a , b ] , then the remainder E n ( f ) in (3.5.2) can be expanded in the form … For C functions Gauss quadrature can be very efficient. …
5: 6.16 Mathematical Applications
It occurs with Fourier-series expansions of all piecewise continuous functions. … …
6: 1.8 Fourier Series
Let f ( x ) be an absolutely integrable function of period 2 π , and continuous except at a finite number of points in any bounded interval. … If a n and b n are the Fourier coefficients of a piecewise continuous function f ( x ) on [ 0 , 2 π ] , then … If a function f ( x ) C 2 [ 0 , 2 π ] is periodic, with period 2 π , then the series obtained by differentiating the Fourier series for f ( x ) term by term converges at every point to f ( x ) . …
7: 1.17 Integral and Series Representations of the Dirac Delta
1.17.2 - δ ( x - a ) ϕ ( x ) d x = ϕ ( a ) , a ,
From the mathematical standpoint the left-hand side of (1.17.2) can be interpreted as a generalized integral in the sense that
1.17.3 lim n - δ n ( x - a ) ϕ ( x ) d x = ϕ ( a ) ,
1.17.6 lim n n π - e - n ( x - a ) 2 ϕ ( x ) d x = ϕ ( a ) ,
1.17.19 lim n - π π δ n ( x - a ) ϕ ( x ) d x = ϕ ( a ) ,
8: 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 ) . … If q ( x ) is C on the closure of ( a , b ) , then the discretized form (3.7.13) of the differential equation can be used. …
9: 1.5 Calculus of Two or More Variables
§1.5(i) Partial Derivatives
A function f ( x , y ) is continuous at a point ( a , b ) if … A function is continuous on a point set D if it is continuous at all points of D . 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 . …
10: 2.1 Definitions and Elementary Properties
For example, suppose f ( x ) is continuous and f ( x ) x ν as x + in , where ν ( ) is a constant. …
2.1.11 x f ( t ) d t - x ν + 1 ν + 1 , ν < - 1 ,
2.1.12 f ( x ) d x { a constant, ν < - 1 , ln x , ν = - 1 , x ν + 1 / ( ν + 1 ) , ν > - 1 .