Examples
Given a topological space X, we can equip the space of bounded real or complex-valued functions over X with the uniform norm topology. Then uniform convergence simply means convergence in the uniform norm topology.
The sequence with converges pointwise but not uniformly:
In this example one can easily see that pointwise convergence does not preserve differentiability or continuity. While each function of the sequence is smooth, that is to say that for all n, the limit is not even continuous.
Read more about this topic: Uniform Convergence
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