uniform convergence
Pointwise convergence lets each point set its own pace; uniform convergence demands they all march in step. Picture a tube of half-width epsilon drawn around the limit function f. Uniform convergence says: from some index N onward, the ENTIRE graph of f_n lies inside that tube — no point of the domain is allowed to stick out. One bound, one N, the whole domain at once.
Precisely: f_n converges uniformly to f on E if for every epsilon > 0 there exists N (depending on epsilon ALONE, not on x) such that |f_n(x) - f(x)| < epsilon for all n >= N and all x in E. Equivalently, the supremum sup over x of |f_n(x) - f(x)| tends to 0 as n -> infinity. That single number measuring the worst-case gap is the sup norm of f_n - f.
Uniform convergence is the workhorse hypothesis of analysis because it transmits good behavior to the limit: a uniform limit of continuous functions is continuous, you may integrate term by term, and (with an extra hypothesis on derivatives) differentiate term by term. The price is that it is genuinely stronger than pointwise convergence and can fail even when pointwise convergence holds perfectly well.
Take f_n(x) = x^n on [0, 1]. The pointwise limit is discontinuous, so it CANNOT be a uniform limit of continuous functions; indeed sup over [0,1) of |x^n - 0| = 1 for every n, never shrinking. But on [0, a] with a < 1 fixed, sup |x^n| = a^n -> 0, so f_n -> 0 uniformly there. Same sequence, uniform on a smaller set, not on the full one.
Uniformity depends on the domain: x^n is uniform on [0,a], a<1, but not on [0,1).
Quick failure test: if sup over x of |f_n(x) - f(x)| does not go to 0, convergence is not uniform. This single computation settles most textbook examples instantly.