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Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function. Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.
For example, the gamma function is a function that satisfies the functional equation (+) = and the initial value () = There are many functions that satisfy these conditions, but the gamma function is the unique one that is meromorphic in the whole complex plane, and logarithmically convex for x real and positive ( Bohr–Mollerup theorem ).
The image of a function f(x 1, x 2, …, x n) is the set of all values of f when the n-tuple (x 1, x 2, …, x n) runs in the whole domain of f.For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either an interval or a single value.
There is a larger class of number-theoretic functions that do not fit this definition, for example, the prime-counting functions. This article provides links to functions of both classes. An example of an arithmetic function is the divisor function whose value at a positive integer n is equal to the number of divisors of n.
The Dirichlet function is not Riemann-integrable on any segment of despite being bounded because the set of its discontinuity points is not negligible (for the Lebesgue measure). The Dirichlet function provides a counterexample showing that the monotone convergence theorem is not true in the context of the Riemann integral.
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If a function is bijective (and so possesses an inverse function), then negative iterates correspond to function inverses and their compositions. For example, f −1 (x) is the normal inverse of f, while f −2 (x) is the inverse composed with itself, i.e. f −2 (x) = f −1 (f −1 (x)).
SPOILERS BELOW—do not scroll any further if you don't want the answer revealed. The New York Times. Today's Wordle Answer for #1304 on Monday, January 13, 2025.
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