The above mentioned theorem, stating that the uniform limit of continuous functions is continuous, remains correct in these settings.

Consider for instance with uniform limit 0, but the derivatives do not approach 0.

The uniform limit of this sequence then played precisely the same role as Bolzano's "limit point".

The function "h" is the uniform limit on compacta of the normalized iterates .

The definition allows only for certain kind of singularities, so that existence, uniqueness, and stability under uniform limits, hold for a large class of equations.

But the police said a uniform limit would allow them to write more speeding tickets, deterring speeders.

The uniform limit of a convergent sequence of harmonic functions is still harmonic.

Even this does not give us what we want, because the Riemann integral no longer commutes with uniform limits.

The Senate, to its credit, voted to set a uniform $125,000 limit on the value of a house that can be shielded.

Note also that the limit function must be continuous, since a uniform limit of continuous functions is necessarily continuous.