Let X be a set. A topology on X is acollection \(\tau \)of subsets of X, for which one of these does not hold: Posted on:
A set \(\bigcup\) is open in the meric topology induced by d if and only for each x\(\epsilon\bigcup\), there exist \(\epsilon> 0\) such that Posted on:
Let \(\mathbb R\) be with the usual standard topology and let A \subsets \mathbb R\).Then A is open in \(\mathbb R\) if there exists an interval I such that I\subset A. For a,b\(\epsilon\mathbb R, I = Posted on:
A metric on a set X with a function d : X x X \(\rightarrow\mathbb R\) holds for all but one property in the following: Posted on:
Let \(\phi(x)=\left\|\begin{array}{rcl}f(x)&g(x)&h(x)\\-f(a)&g(a)&h(a)\\f(b)&g(b)&h(b)\end{array}\right|\), \(g(x)=x\) and \(h(x)\) for all \(x\ina,b\), to deduce the Lagrange’s mean value theorem from the Generalized mean value theorem we must obtain the determinant Posted on:
Applying Cauchy’s mean value theorem to the function \(f\) and \(g\) defined as \(f(x)=x^2\) and \(g(x)=x\) for all \(x\ina,b\), gives Posted on:
The \(n\)th derivative \(f^{(n)} of the function \(f: \mathbb{R}\rightarrow \mathbb{R}\) defined by \(f(x)=|x|\) for all \(x\in\mathbb{R}\) is Posted on: