5 counterexamples every calculus student should know

You can play around with these counterexamples in this MAPLE LEARN document: My thanks to Maple Learn for sponsoring today's video. Claim 1: Discontinuities are isolated Counterexample: The dirichlet function (1 for rationals, 0 for irrationals) is discontinuous everywhere Claim 2: The derivative of a differentiable function is continuous Counterexample: x^2sin(1/x) when x is nonzero, 0 when x=0 Claim 3: A positive derivative at a point implies the function is increasing on some neighbourhood of the point Counterexample: x/2 x^2 sin(1/x) (and 0 when x=0) Claim 4: If a function has a limit at infinity and is differentiable, then it's derivative has a limit at infinity. Counterexample: sin(x^2)/x Claim 4: If f(x) is the limit of a sequence of continuous function f_n(x), then f(x) is also continuous Counterexample: x^n 0:00 The D