I've learned that function $f:\mathbb{R}^n\rightarrow \mathbb{R}^p$, at $x \in \mathbb {R}^n$, $f(x)$ is Gateaux differentiable, then exits a linear operator $\lambda (x):\mathbb{R}^n\rightarrow \mathbb{R}^p$ such that $D_uf(x)=\lim_{t \rightarrow 0}{\frac{f(x+tu)-f(x)}{t}}=\lambda(x)(u),\forall u \in \mathbb{R}^n$. If $\lambda(x)$ is continous, $f(x)$ is Frechet differentiable.   I want to find a counterexample that $f(x)$ is Gateaux differentiable and $\lambda(x)$ is linear, not continous, but it's hard for me, help me, thank you!
Let $V,W$ be nonzero normed spaces over $\mathbb{K}$ such that $V$ is finite-dimensional.  Let $E$ open in $\mathbb{K}$ and $p\in E$.  Let $f:E\rightarrow W$ be Gâteaux-differentiable at $p$.  Is $f$ necessarily Fréchet-differentiable at $p$ in this case?  I think this is not true in general, but cannot find a counterexample. What would be a counterexample?
It seems that most people who use the ignored tags feature end up with a fairly long list of those tags.  Although , it is sometimes necessary to add new ignored tags manually (typically when adding some-tag*).  It would be a lot easier if the add box was above the long list of ignored tags rather than below. I suppose the same would apply to the favorites tags for consistency's sake. Seem to have lost 4 Silver badges and 10 Bronze badges How does Newtonian mechanics explain why orbiting objects do not fall to the object they are orbiting? The force of gravity is constantly being applied to an orbiting object.  And therefore the object is constantly accelerating.  Why doesn't gravity eventually "win" over the object's momentum, like a force such as friction eventually slows down a car that runs out of gas?  I understand (I think) how relativity explains it, but how does Newtonian mechanics explain it? Asking users to accept answer when they comment that it answers their question What is a mixin, and why are they useful? I took some shots recently and I noticed that I'm getting a lot of little black dots in my images.  I cleaned my lens thoroughly with the Nikon camelhair brush and retook the pictures but once again, the dots appear.  I'm shooting with a Nikon D7000 and using a starter lens. I've attached an example (notice the black spots on the chairs, on the ceiling, etc). Is a contradiction enough to prove a set equality to $\varnothing$? In respect to a larger proof I need to prove that $(n-m) \mid (n^r - m^r) $ (where $\mid$ means divides, i.e., $a \mid b$ means that $b$ modulus $a$ = $0$). I have played around with this for a while now but to no avail and would be grateful if you could help me. Evaluating $\int\sqrt{\frac{1-x^2}{1+x^2}}\mathrm dx$ Evaluating $$\int\sqrt{\frac{1-x^2}{1+x^2}}\mathrm dx$$   I had read the , but it doesn't work. Detect edits to add [SOLVED] or [RESOLVED] to the title and direct the user to add an answer "Geometric" or "Geometrical"? I have read the excellent answers to  - my question is specifically about usage.   Is there a best practice for deciding between the variants geometric and geometrical in writing? A  shows that geometrical was more popular until about 1965, when they switched. What grammatical rules, if any, would be relevant here? Limiting double to 3 decimal places Filesystem owner changed by mistake I ran this command on my ubuntu 12.04 by mistake:  chown someuser / -R   On RHEL it would be fixed by this command:  rpm --setugids -a   Is there anyway to fix it in ubuntu server? UPDATE statement to re-assign a column value according to a numeric formula Prove that the product of a non-zero rational and irrational number is irrational. Could you please confirm if this proof is correct?  Theorem: If $q \neq 0$ is rational and $y$ is irrational, then $qy$ is irrational.  Proof: Proof by contradiction, we assume that $qy$ is rational. Therefore $qy=\frac{a}{b}$ for integers $a$, $b \neq 0$. Since $q$ is rational, we have $\frac{x}{z}y=\frac{a}{b}$ for integers $x \neq 0$, $z \neq 0$. Therefore, $xy = a$, and $y=\frac{a}{x}$. Since both $a$ and $x$ are integers, $y$ is rational, leading to a contradiction. How do I reset a lost password (using recovery mode requires me to type the password)? How to use AT commands in Android by 'root' the phone We have various Android devices from Spike, Samsung, Micromax, etc. How can we issue AT commands from Android to the modem? We searched in Google Play but did not find any app related to this.   suggests:     [F]irst you have to root the phone...`    How can we do this? Derivative of the product of operators and Derivative of exponential I'm asked to show that  $$\frac{d(\hat{A}\hat{B})}{d\lambda} ~=~ \frac{d\hat{A}}{d\lambda}\hat{B} + \hat{A}\frac{d\hat{b}}{d\lambda}$$ With $\lambda$ a continuous parameter. Should I use the definition $$\frac{d\hat{A}}{d\lambda} ~=~ \lim_{\epsilon \to 0} \frac{\hat{A}(\lambda + \epsilon) - \hat{A}(\lambda)}{\epsilon}$$ applied to $\hat{A}\hat{B}$ like $$\frac{d(\hat{A}\hat{B})}{d\lambda} ~=~ \lim_{\epsilon \to 0} \frac{\hat{A}(\lambda + \epsilon)\hat{B}(\lambda + \epsilon) - \hat{A}(\lambda)\hat{B}(\lambda)}{\epsilon}$$ and do some algebra to get the RHS of the first equation, or I'm missing something? Another interesting derivative to pay attention to is: $$\frac{d}{d\lambda}\exp(\hat{A}(\lambda) )~?$$ Where to upload too big files? I have a big HTML file that causes no errors in most of the browsers. However, when I open it with, let's say, Google Chrome, the program shuts down and I cannot trace why this is happening.   This HTML file is quite big and cannot be posted to a question in SO, but it should be ran as a whole as I cannot trace where the problem is. What is/are the most appropriate file upload website/s to upload and paste the link into the question? How to overwrite an excel application without prompting the users