Ff Continuous Preimage of Closed Set
Preimage of closed set is closed implies function is continuous
- Thread starter a6623
- Start date
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a6623 Asks: Preimage of closed set is closed implies function is continuous
Let function $f$ be defined from $D \to \mathbb{R}$, $D \subset R$ closed. If the preimage of every closed set $A$ is closed, then $f$ is continuous.
I was trying to use the standard definition of continuity, so take arbitrary $x_0 \in D$ and show that $\forall \frac 1 m, \exists \frac 1 n. |x - x_0| < \frac 1 n \implies |f(x) - f(x_0) | < \frac 1 m$.
Take arbitrary $\frac 1 m$. Let $A$ be the closed set ranging from $[f(x_0 - \frac 1 m), f(x_0 + \frac 1 m)]$ and hence the closed preimage $f^{-1}(A)$ contains $x_0$ as well and also ranges over $[x_0 - \frac 1 m, x_0 + \frac 1 m]$.
Then we know that any $x \in f^{-1}(A)$ satisfies $|x - x_0| < \frac 1 m$ and it also follows that $f(x) \in A$ by definition of preimage, so $|f(x) - f(x_0)| < \frac 1 m$, showing convergence.
Does this proof work?
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I just notice than some mold has grown in the rear windows of my car. I don't think there is any leaks or anything, but every time temps go over zero the environment go incredible humid because all the snow around thawing.
I've asked a local auto hardware store if they have anything for mold and they've recommended me this.
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- In another book I saw that an ensemble can also be defined as a collection of a very large number of assemblies which are essentially independent of one another but which have been made macroscopically as identical as possible.
Now my doubt is that the element in the 1 case is same as assembly in case 2 or different? How they differ from one another?
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I have done some research to find the answer to this, and I am yet to find an answer. My personal and initial speculations were that the phenomenon was related to muscle spasms, however I highly doubt that clenched muscles would produce such a result.
Some of the material I have come across, have mentioned that the phenomenon is restricted to only AC power as opposed to DC. Whilst interesting, exactly how electric power is converted to kinetic energy when an individual touches a live wire or socket still remains unknown to me. To summarise:
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imida k Asks: How do I know which kernel is handling which .nb file?
OS : window 10
Mathematica : v12.2
By my mistake, I opened another mathematica kernel, not knowing a kernel was already running.
I created a lot of .nb files and worked on them in another kernel, while one kernel was already running with many working .nb.
All .nb are still open.
In this state, how can I know which kernel is handling which .nb file?
I guess that there exist ID(a serial?) concept for both kernel and working .nb file.
And I guess that by executing some command on a working .nb, it is possible to figure out the ID of .nb file and the ID of kernel which handles the working .nb.
Source: https://solveforum.com/forums/threads/preimage-of-closed-set-is-closed-implies-function-is-continuous.239818/
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