Real analysis Limits and continuous functions November 26    3

Suppose that $f:\mathbb{R}\to \mathbb{R}$ is continuous on $\mathbb{R}$ and that $$ \lim_{x\to -\infty} f(x)=\lim_{x\to +\infty} f(x) =k$$ Prove that $f$ is bounded and if there exist a point $x_0 \in\mathbb{R}$ such that $f(x_0)>k$, then $f$ attains

Continuous Extension Theorem in real analysis November 26    2

How to prove: if $f: (a,b) \rightarrow R$ is uniformly continuous, then $f$ can be extended to a continuous function $F: [a,b] \rightarrow R$.It's suffice to show that $f$ can be extended to a continuous function $G: [a,b) \rightarrow R$.I saw simila

question about real analysis concerning inequality November 26    2

Let $\epsilon > 0$ be given. Suppose we have that $$a - \epsilon < F(x) < a + \epsilon$$Does it follow that $a - \epsilon < F(x) \leq a $ ??strip away the irrelevant context, and the question becomes: if $c \gt 0$ and $a \lt b+c$ does it follo

Chain rule of multivarible function. November 26    2

If $u=x^{4}f(y/x,z/x)$ then I have to show that $xu_{x}+yu_{y}+zu_{z}=4u.$ How to prove it? $u_{x}=4x^{3}f(y/x,z/x)+x^{4}f_{p}p_{x}+f_{q}q_{y}$ if $p=y/x,q=z/x$. Now how to proceed further. Please help. Thanks.It is just careful expansion: $$ t = x u

When chain rule does not apply November 26    1

Is it possible for $\frac{d}{dt}f(g(t))$ to exist AND be continuous, but not have it equal to $f'(g(t))g'(t)$, so the chain rule doesn't work?Assume that $g'(t)$ exists but it may not be continuous. Does chain rule require strictly $C^1$?I ask becaus

problem uunderstanding chain rule proof step November 26    1

I'm trying to understand the proof of the chain rule explained in this page: don't understand why f (x+h) = f (x) + u (x, h).I really don't get what u is and it dont allows me to uderstand the rest

contradiction to chain rule. November 26    1

As the chain rule states:If $f(u)$ is differentiable at point $u=g(x)$, and $g(x)$ is differentiable at $x$, then the composite function $(f\circ g)(x)=f(g(x))$ is differentiable at $x$ , and $$(f\circ g)'(x)=f'(g(x)).g'(x).$$ Then the question says:

Using the Chain Rule and Product Rule November 26    1

Find $dy/dx: y=x^3(2x-5)^4$. I have been working on this problem for a few hours. I get the problem to $$y'=x^3*(4)*(2x-5)^3*(2)+(2x-5)^4*(3x)^2$$ but then I do not know what to do next. Any help would be appreciated.Everything looks right EXCEPT for

Chain rule error November 26    1

Find $\frac{ \partial ^2 f}{ \partial x ^2}$ where $f(x,y,z)=h(r)$ in $R^3$ except $(0,0,0)$ and $r$ is the usual radius. Attempt: see here$\dfrac{\partial f}{\partial x} = \dfrac{\operatorname{d} h}{\operatorname{d} r}\dfrac{\partial r}{\partial x}$

Intuition behind functional dependence November 26    3

What is the intuition behind functional independence ?(This is defined in the following way: Let $k\leq n$. The $C^1$ functions $F_1,\ldots,F_k:\mathbb{R}^n\rightarrow \mathbb{R}$ are functionally independent if the matrix whose columns are the gradi

Intuition behind the Divergence of series November 26    4

We know that the series is $\sum_{n=1}^ \infty \frac{1}{n}$ diverges. But when we think intuitively, the sum of the series will grow very slowly after some stage, then how can we say that it diverges. How the series $\sum_{n=1}^ \infty \frac{1}{n^2}$

Very basic derivative question November 26    1

Why is the derivative of $e^x + e^{-x}$ equal to $e^x - e^{-x}$ ?By the chain rule, the derivative of $e^{f(x)}$ is $f'(x)e^{f(x)}$. Hence, $$\frac{d}{dx}[e^{x} + e^{-x}] = \left[\frac{d}{dx}(x)\right]\cdot e^{x} + \left[\frac{d}{dx}(-x)\right]\cdot

Contraction Mapping question November 26    2

Let X be the set of continuous real valued functions defined on $[0,\frac{1}{2}]$ with the metric $d(f,g):=\sup_{x\in[0,\frac{1}{2}]} |f(x)-g(x)|$.Define the map $\theta:X\rightarrow X$ such that $$\theta (f)(x)=\int_{0}^{x} \frac{1}{1+f(t)^2} dt$$.

Proof of an application of the contraction mapping theorem to differential equations November 26    1

Proof of an application of the contraction mapping theorem to differential equations
Please consider the theorem below together with the first part of its proof.1) Why is M closed?2) Why is M complete?3) Why is the final integral a continuous function? (The curvy C denotes t ...

You Might Also Like