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[2] This is one of the few situations in mathematics where pointwise convergence implies uniform convergence; the key is the greater control implied by the monotonicity. The limit function must be continuous, since a uniform limit of continuous functions is necessarily continuous.
Terms like piecewise linear, piecewise smooth, piecewise continuous, and others are very common. The meaning of a function being piecewise P {\displaystyle P} , for a property P {\displaystyle P} is roughly that the domain of the function can be partitioned into pieces on which the property P {\displaystyle P} holds, but is used slightly ...
Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of infinitesimals ", it has two major branches, differential calculus and integral calculus .
Plot of Weierstrass function over the interval [−2, 2]. Like some other fractals , the function exhibits self-similarity : every zoom (red circle) is similar to the global plot. In mathematics , the Weierstrass function , named after its discoverer, Karl Weierstrass , is an example of a real-valued function that is continuous everywhere but ...
the book is organized into the following chapters. i. real variables; ii. functions of real variables; iii. complex numbers; iv. limits of functions of a positive integral variable; v. limits of functions of a continuous variable. continuous and discontinuous functions; vi. derivatives and integrals; vii. additional theorems in the differential ...
Supreme Court Chief Justice John Roberts slammed what he described as “dangerous” talk by some officials about ignoring federal court rulings, using an annual report weeks before President ...
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A Florida man accused of a hate crime for killing a gay man is asking a judge to dismiss the charges, saying he acted in self-defense. Gerald Radford testified on Friday that he feared for his own ...