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==Continuous nowhere differentiable functions==
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Each statement of a syllogism is one of 4 types, as follows:
  
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Other mathematicians of the second half of the 19th century shared Hermite's opinion, fearing that similar investigations into the foundations of mathematics would lead to harmful results.<ref>Jarnik p. 41</ref>
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As late as 1920, Jasek is said to have created a "sensation" when he revealed Bolzano's example of a continuous function that is neither monotone in any interval nor has a finite derivative at the points of a certain everywhere dense set. It has been pointed out that Bolzano's function is actually nowhere differentiable, though he neither claimed nor proved this. Bolzano discovered/invented this function about 1830.<ref>Jarnik p. 37-38</ref>
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! Type !! Statement !! Alternative
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More than 30 years after Bolzano's result, the following example by Weierstrass of a continuous function which has a derivative at no point was published:<ref>P. du Bois-Reymond cited in Jarnik</ref><ref>Schultz</ref>
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| style="text-align: center;" | '''A''' || '''All''' $A$ '''are''' $B$ ||
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:$\displaystyle f(x) = \sum_{n=1}^\infty b^n cos(a^n \pi x)$,
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where $0 < b < 1$, $b$ is positive odd integer, and $ab > 1+3\pi/2$
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| style="text-align: center;" | '''I''' || '''Some''' $A$ '''are''' $B$ ||
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A turn of the century address to the American Mathematical Society summarized the situations ....
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| style="text-align: center;" | '''E''' || '''No''' $A$ '''are''' $B$ || (= '''All''' $A$ '''are not''' $B$)
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==Notes==
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| style="text-align: center;" | '''O''' || '''Not All''' $A$ '''are''' $B$ || (= '''Some''' $A$ '''are not''' $B$)
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<references/>
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==Primary sources==
 
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* duBois-Reymond, P. (1875) ''Journal fur die reine und angewandte Mathematik'', 79 pp. 29-31.
 
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==References==
 

Latest revision as of 13:39, 14 October 2015

Each statement of a syllogism is one of 4 types, as follows:

Type Statement Alternative
A All $A$ are $B$
I Some $A$ are $B$
E No $A$ are $B$ (= All $A$ are not $B$)
O Not All $A$ are $B$ (= Some $A$ are not $B$)
How to Cite This Entry:
Whayes43. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whayes43&oldid=32239
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