A book that I am reading claims the following about the function $ g(p) = 0.5 p^{-0.2} + 0.5 p^{-0.5} $ (which is a demand function):
Formal arguments based on the Intermediate Value Theorem and the Implicit Function Theorem would establish that the inverse demand function is well-defined, continuous and strictly decreasing.
I have done some reading on the Intermediate Value Theorem and the Implicit Function Theorem. I understand what the Intermediate Value Theorem is about, but I am not so sure if I have understood the Implicit Function Theorem.
My question is: What might be those formal arguments based on these two theorems, and how do they show that the inverse function is well-defined, continuous and strictly decreasing?
For all $ p \in (0,\infty) $, observe that we have $ \dfrac{0.5}{\sqrt[5]{p}} + \dfrac{0.5}{\sqrt{p}} \in (0,\infty) $. We can thus formally define a function $ g: (0,\infty) \to (0,\infty) $ by $$ \forall p \in (0,\infty): \quad g(p) \stackrel{\text{def}}{=} \frac{0.5}{\sqrt[5]{p}} + \frac{0.5}{\sqrt{p}}. $$
Elucidating the properties of $ g $
$ g $ is continuous on $ (0,\infty) $, as it is a sum of continuous functions from $ (0,\infty) $ to $ (0,\infty) $.
$ g $ is strictly decreasing on $ (0,\infty) $, as it is a sum of strictly decreasing functions from $ (0,\infty) $ to $ (0,\infty) $. This proves that $ g $ is $ 1 $-$ 1 $.
Note that $ \displaystyle \lim_{p \to 0^{+}} g(p) = \infty $ and $ \displaystyle \lim_{p \to \infty} g(p) = 0 $. Hence, by the Intermediate Value Theorem, for every $ y \in (0,\infty) $, we can find a $ p \in (0,\infty) $ such that $ y = g(p) $. This proves that $ g $ is onto.
As $ g $ is $ 1 $-$ 1 $ and onto, $ g^{-1}: (0,\infty) \to (0,\infty) $ exists.
$ \color{darkgreen}{\text{Proof that $ g^{-1} $ is strictly decreasing:}} $
$ \color{darkgreen}{\text{Proof that $ g^{-1} $ is continuous:}} $