Proving Differentiability of f given g'(x) < 0 $\forall$ x

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The discussion centers on proving that if a real-valued function g is defined on the real numbers and its derivative g'(x) is less than zero for all x, then there cannot exist a differentiable function f: R → R such that the composition f(f(x)) equals g(x). This conclusion is drawn from the properties of monotonic functions and the implications of g being strictly decreasing due to its negative derivative.

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  • Basic concepts of differentiability in calculus
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Suppose the real valued g is defined on \mathbb{R} and g&#039;(x) &lt; 0 for every real x. Prove there's no differentiable f: R \rightarrow R such that f \circ f = g.
 
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