Is f(A) Lebesgue measurable when A is lebesgue measurable and f is a function of the class C1?

Question

Let A be a Lebesgue measurable set. Let f: $\mathbb{R} \rightarrow \mathbb{R}$ be a function of the class $C^1$;

Is this true that f(A) is lebesgue measurable?

I know that this is true when f is injective.

But I don't know that when f is not injective.

Could you teach me it is true or not?

Sorry for my poor English.