Рет қаралды 98,275
You can play around with these counterexamples in this MAPLE LEARN document: learn.maplesoft.com/d/AGHGFLK... 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
►beautifulequations.net/pages/...
COURSE PLAYLISTS:
►DISCRETE MATH: • Discrete Math (Full Co...
►LINEAR ALGEBRA: • Linear Algebra (Full C...
►CALCULUS I: • Calculus I (Limits, De...
► CALCULUS II: • Calculus II (Integrati...
►MULTIVARIABLE CALCULUS (Calc III): • Calculus III: Multivar...
►VECTOR CALCULUS (Calc IV) • Calculus IV: Vector Ca...
►DIFFERENTIAL EQUATIONS: • Ordinary Differential ...
►LAPLACE TRANSFORM: • Laplace Transforms and...
►GAME THEORY: • Game Theory
OTHER PLAYLISTS:
► Learning Math Series
• 5 Tips To Make Math Pr...
►Cool Math Series:
• Cool Math Series
BECOME A MEMBER:
►Join: / @drtrefor
MATH BOOKS I LOVE (affilliate link):
► www.amazon.com/shop/treforbazett
SOCIALS:
►Twitter (math based): / treforbazett
►Instagram (photography based): / treforphotography