uniform convergence of integral

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

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

$F(t):={\displaystyle {\int }_a^b}f(x,t)𝑑x=\underset{u\to b-}{lim}{\displaystyle {\int }_a^u}f(x,t)𝑑x$(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 $\epsilon$ there is a value $x_{\epsilon }\in [a,b]$  such that

when  .