Dr. Vogel's Gallery of Calculus Pathologies

Thomas I. Vogel

• Introduction
• First semester calculus
  • A function which is continuous at only one point.
  • A function which is differentiable at a single point, and discontinuous everywhere else.
  • A function with a derivative defined for all x, but whose derivative is discontinuous.
  • A function continuous at all irrationals, discontinuous at all rationals.
  • A continuous, nowhere differentiable function.
• Second semester calculus
  • A divergent alternating series whose terms go to zero
  • The commutative law doesn't hold for certain series
  • A function which doesn't equal its Taylor series, part 1
  • A function which doesn't equal its Taylor series, part 2
• Third semester calculus
  • A non-differentiable function with partial derivatives everywhere.
  • A function without a limit, although limits exist along all lines
  • A function with a single critical point, which is a local minimum but not a global minimum
  • A non-differentiable function with all directional derivatives
  • A function whose mixed partials are unequal
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