Sequences and series of functions
General · Mathematics
Study notes
Q: Consider fₙ(x) = xⁿ on [0,1]. Pointwise limit? Continuous? x < 1: xⁿ → 0. x = 1: 1ⁿ = 1. Limit: f(x) = 0 (x<1), 1 (x=1) - DISCONTINUOUS! Each fₙ continuous, limit not: convergence NOT uniform! (Uniform limit of continuous is continuous!)