The inversion you are speaking of is subject/auxiliary inversion, with deletion of if, in the condition clauses of conditional constructions:
If I had continued to advance ... ⇨ Had I continued to advance ...
The use is very similar to subject/auxiliary inversion with questions, but much more limited.
Nowadays you may invert only a small number of verbs, and these only in the ‘subjunctive’ past forms: had, were, should and could.
If I should continue to advance ... ⇨ okShould I continue to advance ...
If I were to continue to advance ... ⇨ okWere I to continue to advance ...
Inversion with other auxiliaries was common at one time—down to the early years of the 20th Century—but is no longer permitted, unless you are deliberately emulating old-fashioned speech:
If I might suggest that ... ⇨ ∗Might I suggest that ...
Could is borderline: you will still encounter Could I for If I could, but this use is rapidly vanishing.
Just as with questions, were may be inverted even if it is used as a lexical (non-auxiliary) verb:
If I were a rich man ... ⇨ okWere I a rich man...
But inversion of lexical HAVE (that is, non-auxiliary HAVE), although it was common down to the early part of the 20th Century, is no longer permitted unless you are deliberately evoking old use:
If I had a hammer ... ⇨ ∗Had I a hammer ...
Inverted had, were and could are understood as irrealis (unreal) or counterfactual in mode. Should, however, has realis, indicative (albeit tentative) significance, just as it does in uninverted position:
If I go to town I will bring you back a present.
If I should go to town I will bring you back a present.
Should I go to town I will bring you back a present.
All of these inverted constructions are markedly formal. They are rarely encountered in conversation, and they are never obligatory. You may omit them entirely from your speech and writing, in all registers, and they will never be missed.