# 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.

Title uniform convergence of integral UniformConvergenceOfIntegral 2013-03-22 14:40:30 2013-03-22 14:40:30 pahio (2872) pahio (2872) 13 pahio (2872) Definition msc 26A42 SumFunctionOfSeries ConvergenceOfIntegrals integral converging uniformly uniformly converging integral