Purpose for the conditions of Link Function I am studying GLM at the moment and have a few questions regarding link functions.

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*Why are the conditions of the link function to be smooth monotonic function? What properties are preserved by having such conditions?


*Furthermore, a lot of material that I am reading always takes the inverse of the link function $g$, but since the conditions are monotonic, an inverse is not guaranteed. So what happens if the link function does not have an inverse?
Any clarification will be much appreciated!


Why are the conditions of the link function to be smooth monotonic function? What properties are preserved by having such conditions?

For a glm the mean $\mu$  is linked to the linear predictor $\eta=\beta^T x$ by the link function $g(\mu)=\eta$. Assuming the link function has an inverse, $g^{-1}=m$, we find that
$$ m(g(\mu))=g^{-1}(g(\mu))=\mu$$  or
$$ \mu =m(\eta) $$ so we call $m$ the mean function. If $g$ where not strictly monotone, so it had no inverse, then the equation
$$ g(\mu)=\eta$$ would have multiple solutions, and so the linear predictor $\eta$ would not specify a unique mean!
How woud you then relate your glm model to data? How wpuld you estimate the parameters $\beta$ if the same observed mean could be equally explained by different parameter values? It would not work.

Furthermore, a lot of material that I am reading always takes the inverse of the link function , but since the conditions are monotonic, an inverse is not guaranteed. So what happens if the link function does not have an inverse?

What is meant is strictly monotonic, and then an inverse always exists. The above paragraph explains what will happen if an inverse does not exist. Therefore, existence of an inverse is part of the definition of an glm.
As for your new question in comments:

What is the purposes of the differentiable condition?

That condition is maybe not as fundamental as is monotonicity, but as soon as you start to develop the likelihood theory for glm's, first and second derivatives start to appear. Maybe one could do without it, but the theory would be more difficult, and the need does not seem to be there. Also, numerical optimization of non-differentiable functions is more difficult. But if you see some application for a non-diff link function, just try!
