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limit function of sequence
Theorem 1.
Let be a sequence of real functions all defined in the interval . This function sequence converges uniformly to the limit function on the interval if and only if
If all functions are continuous in the interval and in all points of the interval, the limit function needs not to be continuous in this interval; example in :
Theorem 2.
If all the functions are continuous and the sequence converges uniformly to a function in the interval , then the limit function is continuous in this interval.
Note. The notion of uniform convergence can be extended to the sequences of complex functions (the interval is replaced with some subset of ). The limit function of a uniformly convergent sequence of continuous functions is continuous in .
Mathematics Subject Classification
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.)40A30 Convergence and divergence of series and sequences of functions
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