Here is a sequence of functions (in blue) converging pointwise to the function (in red).

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Here is another  sequence of functions () which converges pointwise to for  -1<x<1.  All functions in the sequence are continuous everywhere, but the pointwise limit over [0,1]  is discontinuous at 1.

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The sequence converges to for x in [-1,1]; but the derivatives do not converge to the derivative of 0.

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Lets take a look at the maximal distance between the functions in the sequence and their pointwise limit . The green vertical line is indicating the maximal distance. The length of that line (the value of the maximal distance) converges to 0 (hence does converge uniformly to  ). Observe though that the rate at which the max distance (green) converges  to 0 changes substantially as we move towords the point x=0.

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The sequence converges to for x in [0,1]; but the integrals of the functions in the sequence (for x in [0,1]) are all 1 while the integral of is 0.

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Here is the same animation with an addition of the vertical green line of the size equal to the maximal distance and at the place where the maximal distance occurs. The convergence is not uniform because the green line does not shrink to 0 (in fact, it increases unboundedly with n).

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However, if we restrict the domain of the functions to the interval [0.2, 1] (thus avoiding the "bad" point 0), than the convergence is uniform; observe how the maximal distance does go to 0 now.

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© S.Kalajdzievski