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(mathematics) A type of convergence of a sequence of functions { f'ₙ }, in which the speed of convergence of f'ₙ(x) to f(x) does not depend on x.

A sequence of functions ${\displaystyle (f_{n})}$ converges uniformly to ${\displaystyle f}$ when for arbitrary small ${\displaystyle \varepsilon }$ there is an index ${\displaystyle N}$ such that the graph of ${\displaystyle f_{n}}$ is in the ${\displaystyle \varepsilon }$-tube around ${\displaystyle f}$ whenever ${\displaystyle n\geq N.}$

(mathematics) A type of convergence of a sequence of functions { f'ₙ }, in which the speed of convergence of f'ₙ(x) to f(x) does not depend on x.

Counterexample to a strengthening of the uniform convergence theorem, in which pointwise convergence, rather than uniform convergence, is assumed. The continuous green functions ${\displaystyle \sin ^{n}(x)}$ converge to the non-continuous red function. This can happen only if convergence is not uniform.