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[Edwardfaw]

Keymacro allows you to quickly and easily manage all of your RSA keys! It's simple, yet powerful. You can create your own personal keypress sequences to automate any action, save your time and efforts, and have fun at the same time! With a single click, you can change your keys, modify or delete them. Additionally, it is very easy to restore any deleted or modified key.
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Keymacro supports the following languages: English, Portuguese, Italian, French, German, Spanish, and more.
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[Johnnysok]

This Macro allows you to use your stored Certificate's Private Key to digitally sign, and thus encrypt, the communication over a TLS 1.2 connection. The macro can decrypt, and thus decrypt, the TLS communication using your Certificate's Public Key. When this Macro is used, the TLS 1.2 traffic is automatically authenticated by using the digital signature provided by the Macro. The Macro can be created using: CTRL+Shift+M. When it is used: CTRL+M.
Thanks for your request. Unfortunately it's not possible to remove the Macros when the email messages have been sent. I tried to change this and will keep you posted here when I have more information.Q:

Prove that if $f$ is odd then $f(x) = 0$ for $x = \pi$

Prove that if $f$ is odd then $f(x) = 0$ for $x = \pi$
Suppose $f(x) = g(x)$ and $g(x)$ is even. Then $f(x) = -f(-x)$ but $-\pi = \pi$ so we have a contradiction.
Is this proof correct?

A:

There is a much simpler proof. Let $f(x)$ be even, that is,
$$
f(x)=f(-x).
$$
Then
$$
f(\pi)=f(\pi+\pi+\cdots+\pi)=f(0)=0.
$$

A:

You can prove that if $f(x)$ is odd then $f(x)=0$ for $x=\pi$ in many ways, not only by using continuity. Here is a standard proof that's easy to understand.
For $\alpha\in\mathbb R$, let $x_\alpha:=\pi+\alpha$. Then $x_\alpha=\alpha\pi$, so $x_\alpha\to\pi$ and $f(x_\alpha)\to f(\pi)$ as $\alpha\to0$. It follows that
$$
f(\pi) = \lim_{\alpha\to0} f(x_\alpha) = \lim_{\alpha\to0}\lim_{\beta\to0}f(x_\alpha+\beta) = \lim_{\beta\ 70238732e0 floque
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[Edwardfaw]

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[Johnnysok]

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