Answer the following about functions that are continuous and not differentiable, for 10 points each.
[10e] This non-negative function is continuous everywhere but is not differentiable at zero. A vertical bar on each side of the input is used to symbolize this function.
ANSWER: absolute value [or modulus; or magnitude; or L1-norm; prompt on norm]
[10h] The existence of a function that is continuous but nowhere differentiable can be proved by showing that the set of anywhere differentiable functions is meager in the space of continuous functions, then applying this person’s namesake “category theorem.”
ANSWER: René-Louis Baire (“bair”) [accept Baire category theorem]
[10m] The vertical bar notation for absolute value was introduced by this person, who names a continuous but nowhere differentiable function equal to a sum of cosines of exponentially-growing frequency. Bernard Bolzano and this person co-name a theorem showing that convergent subsequences exist for bounded sequences.
ANSWER: Karl Weierstrass [or Karl Theodor Wilhelm Weierstrass; accept Weierstrass function; accept Bolzano–Weierstrass theorem]
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