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Before real numbers are precisely defined, $f(x+y) = f(x)+f(y)$ and $f(xy) = f(x)f(y)$... show $f$ preserves order.
Spivak Calculus, 4th ed., problem 3-17:
If $f(x)=0$ for all $x$, then $f$ satisfies $f(x+y)=f(x)+f(y)$ for all $x$ and $y$, and also $f(x\cdot y ) =f(x)\cdot f(y)$ for all $x$ and $y$. Now suppose ...
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One to one function behaviour
Like in pigeon hole principle , if one set of objects(S1) has more items than others set of objects(S2) and we try to fit that S1 in S2 ( that is mapping the values of S1 to S2 , we end up getting ...
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Change of coordinate codomain from $[-1,1]$ to $[0,1]$
How does one translate coordinates from $[-1,1]$ to $[0,1]$? That is, suppose we have an ordered pair $(x,y)$ which lies between $[-1,1]$ and want to push into the range delimited by $[0,1]$.
A lot ...
