Dr. Vogel's Gallery of Calculus Pathologies
Thomas I. Vogel
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
, 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
About this document ...
Mon May 5 12:53:33 CDT 1997