uniform convergence of integral

Let the function $f(x,\,t)$ be continuous in the domain

a\;\leqq\;x\;<\;b,\quad c\;\leqq\;t\;\leqq\;d,

where $b$ is a real number or $\infty$ , and let the improper integral

\displaystyle F(t)\;:=\,\int_{a}^{b}f(x,\,t)\,dx\;=\;\lim_{u\to b-}\int_{a}^{u% }f(x,\,t)\,dx

(1)

be convergent (http://planetmath.org/ImproperIntegral) in every point $t$ of the interval $[c,\,d]$ . We say that the on the interval $[c,\,d]$ , if for each positive number $\varepsilon$ there is a value $x_{\varepsilon}\in[a,\,b]$ such that

\left|\int_{x}^{b}f(x,\,t)\,dx\right|\;<\;\varepsilon\quad\forall t\in[c,\,d]

when $x_{\varepsilon}\leqq x<b$ .

Title	uniform convergence of integral
Canonical name	UniformConvergenceOfIntegral
Date of creation	2013-03-22 14:40:30
Last modified on	2013-03-22 14:40:30
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	13
Author	pahio (2872)
Entry type	Definition
Classification	msc 26A42
Related topic	SumFunctionOfSeries
Related topic	ConvergenceOfIntegrals
Defines	integral converging uniformly
Defines	uniformly converging integral