Is there an equivalent to the Intermediate Value Theorem for $\mathbb{R}^2$ scalar fields? I am to prove some sort of mean value theorem for double integrals. That is, 

if $f: R \subset \mathbb{R}^2 \rightarrow \mathbb{R}$ is continuous on some rectangle $R$, then there exists $c \in R$ such that $\iint_R f\, dA = f(c) \mu(R)$, where $\mu(R)$ is the area of the rectangle $R$.

My only idea so far is to prove this theorem the same way I proved it for real valued functions of real variables: using the Intermediate Value Theorem thanks to the hypothesis that $f$ is continuous. However I am unsure if such theorem exists for several variables. Is there an equivalent to it?

Yes, here it is! Mean value theorem for several variables and also for vector-valued functions. 
https://en.wikipedia.org/wiki/Mean_value_theorem#Mean_value_theorem_in_several_variables
