pointwise convergence
Imagine a whole army of functions f_1, f_2, f_3, ... and you want to say they settle down to one limit function f. The cheapest way to check is to stand at one input x, freeze it, and watch the numbers f_1(x), f_2(x), f_3(x), ... If those numbers converge to f(x), you score a point — literally a point. Pointwise convergence just means you win at every x, but each x is judged on its own, with no coordination between them.
Precisely: a sequence of functions f_n on a set E converges pointwise to f if for every x in E the sequence of real numbers f_n(x) converges to f(x). Spelled out with quantifiers: for each x and each epsilon > 0 there exists N (which may depend on BOTH x and epsilon) such that |f_n(x) - f(x)| < epsilon for all n >= N. The crucial phrase is 'may depend on x': different points are allowed to converge at wildly different speeds.
The honest caveat that trips everyone up: pointwise convergence preserves almost nothing you care about. Continuous functions can converge pointwise to a discontinuous limit; integrals and derivatives of f_n need not approach those of f. That fragility is exactly why uniform convergence — where one N works for all x at once — was invented.
On [0,1] let f_n(x) = x^n. For x < 1, x^n -> 0; at x = 1, x^n = 1 -> 1. So f_n converges pointwise to the function f that is 0 on [0,1) and 1 at x = 1. Each f_n is continuous, yet the limit jumps — a vivid failure of continuity under mere pointwise convergence.
x^n on [0,1]: continuous terms, discontinuous pointwise limit.
Pointwise convergence is the weakest natural notion: uniform convergence implies it, but not conversely. When you read 'f_n -> f' with no qualifier, context decides which is meant — analysts usually flag uniform convergence explicitly.