Let f be a measurable function and g be a 1-1 function from R to R which has a Lipschitz inverse.

Show that the composition fog is measurable

My idea was to take a set (a,00) , then f^-1(a,00) is measurable call this set C , but i need to prove that g^-1(C) is measurable

Since g^-1 is Lipschitz then it is continuous. Let g^-1 = h , so how can I prove that h(C) is measurable provided that h is Lipschitz, hence continuous ??

Any ideas please ?? Thanks