Merging man & maths https://www.mathcity.org/ Fri, 05 Jun 2026 08:37:46 +0000 FeedCreator 1.8 https://www.mathcity.org/_media/logo.svg https://www.mathcity.org/ https://www.mathcity.org/msc/real_analysis_notes_by_syed_gul_shah/real_number_system Chapter 01 - Real Number System Contents & Summary * Theorem: There is no rational p such that $p^2=2$. * Theorem: Let A be the set of all positive rationals p such that $p^2>2$ and let B consist of all positive rationals p such that $p^2<2$ then A contain no largest member and $x<y$$x<u<y$$x=\sup E$$x>0$$n>0$$y^n=x$$\underline x,\underline y\in \mathbb{R}^n$$\|\underline x^2\|=\underline x\cdot \underline x$$\|\underline x\cdot \underline y\|=\|\underline x\| \|\underline y\|$$\underline … anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:49:59 +0000 https://www.mathcity.org/msc/real_analysis_notes_by_syed_gul_shah/differentiation Chapter 04 - Differentiation * Derivative of a function * Theorem: Let f be defined on [a,b], if f is differentiable at a point $x\in [a,b]$, then f is continuous at x. (Differentiability implies continuity) * Theorem (derivative of sum, product and quotient of two functions)$x\in [a,b]$$f'(x)$$f'(x)=0$$\mathbb{R}^k$$\underline{f}$$x\in (a,b)$$\left|\underline{f}(b)-\underline{f}(a)\right|\le (b-a)\left|\underline{f'}(x)\right|$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:49:57 +0000