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A complex-valued function of a real variable may be defined by relaxing, in the definition of the real-valued functions, the restriction of the codomain to the real numbers, and allowing complex values. If f(x) is such a complex valued function, it may be decomposed as f(x) = g(x) + ih(x), where g and h are real-valued functions. In other words ...
More precisely, whereas a function satisfying an appropriate summability condition defines an element of L p space, in the opposite direction for any f ∈ L p (X) and x ∈ X which is not an atom, the value f(x) is undefined. Though, real-valued L p spaces still have some of the structure described above in § Algebraic structure.
This function is continuous on the closed interval [−r, r] and differentiable in the open interval (−r, r), but not differentiable at the endpoints −r and r. Since f (−r) = f (r), Rolle's theorem applies, and indeed, there is a point where the derivative of f is zero. The theorem applies even when the function cannot be differentiated ...
Assuming that [a − r, a + r] ⊂ I and r < R, all these series converge uniformly on (a − r, a + r). Naturally, in the case of analytic functions one can estimate the remainder term R k ( x ) {\textstyle R_{k}(x)} by the tail of the sequence of the derivatives f′ ( a ) at the center of the expansion, but using complex analysis also ...
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...
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There are two ways to track the effect of changing R to f(R), i.e., to obtain the theory field equations. The first is to use metric formalism and the second is to use the Palatini formalism. [3] While the two formalisms lead to the same field equations for General Relativity, i.e., when f(R) = R, the field equations may differ when f(R) ≠ R.