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The pointwise limit of a sequence of continuous functions may be a discontinuous function, but only if the convergence is not uniform. For example, f ( x ) = lim n → ∞ cos ( π x ) 2 n {\displaystyle f(x)=\lim _{n\to \infty }\cos(\pi x)^{2n}} takes the value 1 {\displaystyle 1} when x {\displaystyle x} is an integer and 0 {\displaystyle ...
A sequence of functions () converges uniformly to when for arbitrary small there is an index such that the graph of is in the -tube around f whenever . The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions () = (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).
The pointwise limit function need not be continuous, even if all functions are continuous, as the animation at the right shows. However, f is continuous if all functions f n {\displaystyle f_{n}} are continuous and the sequence converges uniformly , by the uniform convergence theorem .
The following theorem establishes conditions for the pointwise limit of a sequence of continuous linear maps to be itself continuous. Theorem [ 4 ] — Suppose that h 1 , h 2 , … {\displaystyle h_{1},h_{2},\ldots } is a sequence of continuous linear maps between two topological vector spaces X {\displaystyle X} and Y . {\displaystyle Y.}
This is one of the few situations in mathematics where pointwise convergence implies uniform convergence; the key is the greater control implied by the monotonicity. The limit function must be continuous, since a uniform limit of continuous functions is necessarily continuous.
Then the function f(x) defined as the pointwise limit of f n (x) for x ∈ S \ N and by f(x) = 0 for x ∈ N, is measurable and is the pointwise limit of this modified function sequence. The values of these integrals are not influenced by these changes to the integrands on this μ-null set N , so the theorem continues to hold.
In particular, the limit of an equicontinuous pointwise convergent sequence of continuous functions f n on either metric space or locally compact space [1] is continuous. If, in addition, f n are holomorphic, then the limit is also holomorphic.
In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in measure theory due to Lebesgue and Beppo Levi that says that for sequences of non-negative pointwise-increasing measurable functions (), taking the integral and the supremum can be interchanged with the result being finite if either one is ...