# Questions tagged [smooth-functions]

For questions about infinitely or arbitrarily differentiable (smooth) functions of one or several variables. To be used especially for real-valued functions; for complex-valued functions, the tag holomorphic-functions is more appropriate.

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### Prove that function is convex, first order smoothness,second order smoothness

I have the following function: $$\mathbb{R}^n \ni (x_1, \cdots , x_n) \mapsto \ln( \sum_{k=1}^{n}{exp(x_k)})$$ I know how to prove if a function is convex, but I have trouble with this specific one. ...
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### What's the smoothness of the 4-D spacetime?

If treating General Relativity's 4-D spacetime as a function $f(x,y,z,t)$. Now use it to describe the timeline of a ball, so $f(x,y,z,t)$ will return you a real number that represents how much ...
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### approximate identity- common compact set

let us consider a sequence of smooth functions with compact support $\phi_n$, which approximate Dirac measure at zero, such that $\int_{\mathbb{R}}\phi_n(x)\,dx=1$. Okay, I can consider something like ...
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### If $f$ is a continuous function and $\phi$ is a smooth function with compact support such that $\int\phi\,dx=1$, is $f*\phi$ Lipschitz continuous?

Let us take a continuous function $f$. Let $\phi$ be a smooth function with a compact support, such that $\int \phi(x)\,dx=1$. We consider the following convolution $(f*\phi)(x)=\int f(x-y)\phi(y)\,dy$...
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### Determining the differential of a map defined on a submanifold of $\Bbb R^n$

Let $M=\{(x,y,z,w)\in \Bbb R^4:x^3+y^3+z^3-3xyz=1\}$ and consider the function $f:M\to \Bbb R^2$ defined by $f(x,y,z,w)=(x+y+z+w,w^3+w)$. It is easily checked that $1$ is a regular value of the ...
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### A way to prove that a function is smooth except in zero

So i got this demonstration where i have to show that $E \in C^{\infty}(R^n-\{0\})$. If $\alpha$ is a multi index, $D_{\alpha}= \partial_{x_1}^{\alpha_1} \ldots \partial_{x_n}^{\alpha_n})$, in my ...
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### Show that $C^\infty(\overline{\Omega}) \subseteq C^{0,1} (\overline{\Omega})$

Let $\Omega$ be a bounded, connected, open domain in $\mathbb{R}^d$ with smooth boundary. Denote by $C^{0,1} (\overline{\Omega})$ the space of continuous functions $u$ on $\overline{\Omega}$ such ...
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### Where is $f(x) = |x^2(x+1)|$ differentiable? And where are they $C^1$ and $C^2$?

I'm a maths student taking a real-analysis paper and I'm currently working down my problem sheet. I've been asked the above question. First I define a piece-wise function to describe the absolute ...
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### Can $\frac{\mathrm d}{\mathrm dx}$ increase the support of a function?

Let "support" mean the closed support. Can $\frac{\mathrm d}{\mathrm dx}$ increase the support of a function? That is, is there any $f=f(x)$ in one variable with $\operatorname{supp}(f)$ completely ...
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### How to formally state and show that, for a smooth function from $R^2$ to $R$, the intersection of a contour line with itself is a saddle point?

I am referring to the line from https://en.wikipedia.org/wiki/Saddle_point (second paragraph, last sentence): In terms of contour lines, a saddle point in two dimensions gives rise to a contour ...
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### How to remove singularity from the solution?

Suppose we have $f(x)=\frac{1}{sin(x-x0)cos(x-x0)}$, we know that $f(x)$ is undefined at $x=x0$. Is there any possibility to remove this singularity so that $f(x)$ would be a smooth ...
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I don't know much about differential geometry but as I understand it for a map between manifolds $F : M \to N$ to be $C^1$ around $p$ means that there exists charts $(U,\phi)$ containing $p$ and $(V,\... 1answer 39 views ### function smoothing with exponent I found this statement in a proof and thought it is clear, thinking of the absolute value function as an example, but I can not give a proof. Can someone give me a hint? There is a function$f: R^6 \...
Let $(u_k)_{k\geq 0}$ and $(v_k)_{k\geq 0}$ be two sequences with $u_0 \lt v_0$ and $u_1\gt 0,v_1\gt 0$. My question : is there an $f\in{\mathcal C}^{\infty}([0,1],{\mathbb R})$ with $f^{(k)}(0)=u_k$ ...
Suppose $M$ is a smooth manifold and $f : M \to \mathbb R$ is a continuous function. The function $f$ may not be smooth, but does there exist another smooth structure $M'$ (on the same topological ...