Summary
Keywords
Full Transcript
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 Check out my MATH MERCH line in collaboration with Beautiful Equations ►https://beautifulequations.net/pages/trefor COURSE PLAYLISTS: ►DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS ►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6 ►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m ► CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n ►MULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd ►VECTOR CALCULUS (Calc IV) https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa ►DIFFERENTIAL EQUATIONS: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxde-SlgmWlCmNHroIWtujBw ►LAPLACE TRANSFORM: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1 ►GAME THEORY: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxdzD8KpTHz6_gsw9pPxRFlX OTHER PLAYLISTS: ► Learning Math Series https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw ►Cool Math Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho BECOME A MEMBER: ►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join MATH BOOKS I LOVE (affilliate link): ► https://www.amazon.com/shop/treforbazett SOCIALS: ►Twitter (math based): http://twitter.com/treforbazett ►Instagram (photography based): http://instagram.com/treforphotography
Continue this lesson in the app
Install CourseHive on Android or iOS to keep learning while you move.
