Can I say
..., giving the example of Minnesota, you would probably pay only half of that
instead of
... to give the example of Minnesota, you would probably pay only half of that ?
What is the difference in meaning between "giving" and "to give"? My guess is that "giving" means "if I give the example". Or "to give" is the abbreviated form of "if I were to give"?
You also left out the previous part of the quote, which tells us what the pronoun "that" refers to. My guess is that this is from an article on something like apartment rentals, comparing the rents in different parts of the country.
I also guess you copied the original quote incorrectly. Here's a grammatical version:
In other parts of the country, again, to give the example of Minnesota, you would probably pay only half of that -- maybe $400 or 450 a month.
But to address your question: Yes, you can use "giving" instead of "to give". In many cases the infinitive and the gerund forms of a verb can both be used, although not necessarily in the same way.
Giving an example of another part of the country, Minnesota, where you would probably pay only half of that.
However this grammar is unnecessarily convoluted, and it would be better to keep things simple:
In other parts of the country, like Minnesota, you would only pay half of that, around $400 or $450 per month.
Neither of them sound very good. Instead consider
In another part of the country, for example, Minnesota, you would probably pay only half of that---Maybe $400 or $450 a month
Hard to answer your question about give versus giving because it's not a natural expression in this context. You could possibly use give as follows:
Let me give you an example: in Minnesota you would probably pay only half of that---perhaps $400 or $450 a month.
I can't torture your original sentence into a good one that uses "giving."
