Why is a nondecreasing function bounded on [a,b]?
In the complex plane, sine & cosine are no longer bounded. However, the relation we all know and love still survives.pic.twitter.com/IJmNMHvWVm
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Because it’s a constant function! Haha
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The statement in your tweet is stronger (though still true) than the statement in your LaTeX inequalities.
As it is written there it does not necessarily imply that the cosine and sine are unbounded, for that one would have to add that no constant exists such that |sin(z)|<=C. -
In fact, as the absolute value function on the complex plane in general maps to the half-open real interval [0,infinity), the statement that the absolute value of a well-defined function at a point lies in [0,infinity) follows from the definition of the absolute value function.
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