Let be a sequence of functions, each regular in a region , let for every and in , and let tend to a limit as at a set of points having a limit point inside . Then tends uniformly to a limit in any region bounded by a contour interior to , the limit therefore being an analytic function of .

# Vitali's Convergence Theorem

## See also

Montel's Theorem## Explore with Wolfram|Alpha

## References

Titchmarsh, E. C.*The Theory of Functions, 2nd ed.*Oxford, England: Oxford University Press, p. 168, 1960.

## Referenced on Wolfram|Alpha

Vitali's Convergence Theorem## Cite this as:

Weisstein, Eric W. "Vitali's Convergence Theorem."
From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/VitalisConvergenceTheorem.html