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Continuity, Intermediate Values, and Darboux's Theorem

Can f' exist at a place where it fails to be continuous? A look at the subtleties within Calculus. Intended audience: Moderately strong confidence with Calculus I concepts and calculations. Some results proved. 0:00 - Morning Goat intro theme ( URL:   / morning-goat   ) 1:11 - Absolute value function, point of non-differentiability. (Derivative at c DNE and limit of derivative as x approaches c DNE) 6:00 - If f' is discontinuous at c, does f' necessarily not exist at c? 8:05 - sin(1/x) and intermediate value property 20:00 - Can f' have a discontinuity at c but exist AND satisfy the intermediate value property around c? 21:00 - First attempt at differentiabilty at a discontinuity of f' - x sin(1/x) 26:30 - Second attempt - x^2 sin(1/x) 27:00 - x^2 sin(1/x) - limit of derivative as x approaches 0 34:00 - x^2 sin(1/x) - derivative at 0 39:00 - Darboux's Theorem - IVP for derivatives 41:50 - Examples - Satisfy Darboux's Theorem (or N/A) 47:20 - Proof of Darboux's Theorem (or half of it anyway, the other half is the same idea with routine modifications, or a case reduction argument) 58:25 - Corollaries and related facts (proof omitted) 1:03:00 - Darboux's Theorem implies that absolute value function can't be differentiable at 0 1:07:25 - "Good place to stop" + outro (Morning Goat part 2)