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Dr. Vogel's Gallery of Calculus Pathologies http://www.math.tamu.edu/~tvogel/gallery/gallery.html

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Dr. Vogel's Gallery of Calculus Pathologies http://www.math.tamu.edu/~tvogel/gallery/gallery.html

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Introduction http://www.math.tamu.edu/~tvogel/gallery/node1.html#SECTION00010000000000000000

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A function which is continuous at only one point. http://www.math.tamu.edu/~tvogel/gallery/node3.html#SECTION00021000000000000000

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A function which is differentiable at a single point, and discontinuous everywhere else. http://www.math.tamu.edu/~tvogel/gallery/node4.html#SECTION00022000000000000000

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A function with a derivative defined for all x, but whose derivative is di scontinuous. http://www.math.tamu.edu/~tvogel/gallery/node5.html#SECTION00023000000000000000

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A function with a derivative defined for all x, but whose derivative is di scontinuous. http://www.math.tamu.edu/~tvogel/gallery/node5.html#SECTION00023000000000000000

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A function with a derivative defined for all x, but whose derivative is di scontinuous. http://www.math.tamu.edu/~tvogel/gallery/node5.html#SECTION00023000000000000000

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A function continuous at all irrationals, discontinuous at all rationals. http://www.math.tamu.edu/~tvogel/gallery/node6.html#SECTION00024000000000000000

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A continuous, nowhere differentiable function. http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000

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A continuous, nowhere differentiable function. http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000

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A continuous, nowhere differentiable function. http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000

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A continuous, nowhere differentiable function. http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000

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A continuous, nowhere differentiable function. http://www.math.tamu.edu/~tvogel/gallery/node7.html#SECTION00025000000000000000

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A divergent alternating series whose terms go to zero http://www.math.tamu.edu/~tvogel/gallery/node9.html#SECTION00031000000000000000

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The commutative law doesn't hold for certain series http://www.math.tamu.edu/~tvogel/gallery/node10.html#SECTION00032000000000000000

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The commutative law doesn't hold for certain series http://www.math.tamu.edu/~tvogel/gallery/node10.html#SECTION00032000000000000000

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A function which doesn't equal its Taylor series, part 1 http://www.math.tamu.edu/~tvogel/gallery/node11.html#SECTION00033000000000000000

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A function which doesn't equal its Taylor series, part 2 http://www.math.tamu.edu/~tvogel/gallery/node12.html#SECTION00034000000000000000

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A function which doesn't equal its Taylor series, part 2 http://www.math.tamu.edu/~tvogel/gallery/node12.html#SECTION00034000000000000000

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A non-differentiable function with partial derivatives everywhere. http://www.math.tamu.edu/~tvogel/gallery/node14.html#SECTION00041000000000000000

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A non-differentiable function with partial derivatives everywhere. http://www.math.tamu.edu/~tvogel/gallery/node14.html#SECTION00041000000000000000

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A non-differentiable function with partial derivatives everywhere. http://www.math.tamu.edu/~tvogel/gallery/node14.html#SECTION00041000000000000000

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A function without a limit, although limits exist along all lines http://www.math.tamu.edu/~tvogel/gallery/node15.html#SECTION00042000000000000000

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A function without a limit, although limits exist along all lines http://www.math.tamu.edu/~tvogel/gallery/node15.html#SECTION00042000000000000000

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A function with a single critical point, which is a local minimum but not a global minimum http://www.math.tamu.edu/~tvogel/gallery/node16.html#SECTION00043000000000000000

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A function with a single critical point, which is a local minimum but not a global minimum http://www.math.tamu.edu/~tvogel/gallery/node16.html#SECTION00043000000000000000

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A function with a single critical point, which is a local minimum but not a global minimum http://www.math.tamu.edu/~tvogel/gallery/node16.html#SECTION00043000000000000000

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A diff i bl f i i h l l di i l d i i h // h d / l/ ll / d 17 h l#SECTION00044000000000000000

http://www.math.tamu.edu/~tvogel/gallery/node16.html#SECTION00043000000000000000http://www.math.tamu.edu/~tvogel/gallery/node16.html#SECTION00043000000000000000http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000

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A non-differentiable function with al l di rectional derivatives http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000

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A diff ti bl f ti ith l l di ti l d i ti htt // th t d / t l/ ll / d 17 ht l#SECTION00044000000000000000

http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000

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A non-differentiable function with al l di rectional derivatives http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000

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A f ti h i d ti l l htt // th t d / t l/ ll / d 18 ht l#SECTION00045000000000000000

http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000http://www.math.tamu.edu/~tvogel/gallery/node17.html#SECTION00044000000000000000http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000

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A function whose mixed partials are unequal http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000

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A function whose mixed partials are unequal http://www math tamu edu/ tvogel/gallery/node18 html#SECTION00045000000000000000

http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000

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A function whose mixed partials are unequal http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000

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A function whose mixed partials are unequal http://www math tamu edu/ tvogel/gallery/node18 html#SECTION00045000000000000000

http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000

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A function whose mixed partials are unequal http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000

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About this document http://www math tamu edu/~tvogel/gallery/node19 html#SECTION00050000000000000000

http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000http://www.math.tamu.edu/~tvogel/gallery/node18.html#SECTION00045000000000000000http://www.math.tamu.edu/~tvogel/gallery/node19.html#SECTION00050000000000000000http://www.math.tamu.edu/~tvogel/gallery/node19.html#SECTION00050000000000000000

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About this document ... http://www.math.tamu.edu/~tvogel/gallery/node19.html#SECTION00050000000000000000

1 f 1 22 02 2011 23 08

http://www.math.tamu.edu/~tvogel/gallery/node19.html#SECTION00050000000000000000http://www.math.tamu.edu/~tvogel/gallery/node19.html#SECTION00050000000000000000