 Uniform limit

A function $f:E\to \mathbb{R}$ is the uniform limit of a sequence of functions $(f_n)$ whenEquivalently the supremum error ${\sup }_{x\in E}|f_n(x)-f(x)|$ tends to zero. The same index works for all inputs, including inputs chosen differently for each $n$. A uniform limit of continuous functions is continuous, by approximating both endpoint values with one common approximant.

For $f_n(x)=x^n$ on $[0,a]$, $0<a<1$, the zero function is a uniform limit because the supremum error is $a^n$. On $[0,1]$, the pointwise limit instead equals zero below one and one at one. Its supremum error remains one for every $n$, so there is no uniform limit there. The open supremum below the endpoint need not be attained to obstruct uniform convergence.