You can play around with these counterexamples in this MAPLE LEARN document: https://learn.maplesoft.com/d/AGHGFLKNBGCIBKFRGUHLPHNPBIBLMKFOKSHNFNGSDOIKNNOSCKEGOJNICTDNEOLNKOGHNPHOLTAPOLNNAJDPCFNLASDFFSISMHMN 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 Dirichlet Function
Differentiable doesn't imply continuous derivatives
Positive derivative doesn't imply increasing
Functions and derivatives can have different asymptotics
Limits of sequences don't have to be nice
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