Pointwise vs Uniform Convergence

Real analysis draws a sharp line between two ways a sequence of functions can "converge": pointwise, where each individual point settles down at its own pace, and uniform, where a single tolerance works everywhere on the domain at once. This simulator renders the whole function sequence f_n(x) as a 3D waterfall surface — x across, the index n receding into depth, height equal to f_n(x), and colour equal to the pointwise error against the limit function. An adjustable ε-band traces the limit function through the depth axis, a live sup-norm readout reports the true worst-case error at the current n, and three textbook examples (xⁿ, sin(nx)/n, and the travelling bump n·x·(1−x)ⁿ) let you see uniform and non-uniform convergence side by side, including the classic case where the limit is continuous yet convergence still fails to be uniform.