What is the relation between compactness , connectedness and continuous real-valued functions on $\mathbb{R^n}$, n > 1? What is the relation between compactness, connectedness and continuous real-valued functions on $\mathbb{R}^n$, n > 1?
For example,
1- what is the relation between a compact set and boundedness of every continuous real valued function on it? 
2- If a set on $\mathbb{R}^n$ is compact, is it bounded? and if it is bounded is it compact? 
 A: Suppose $X$ is compact and $f:X\to\mathbb{R}^n$ is continuous. Then $f$ is bounded.
Proof:
Since the continuous image of compact spaces are compact, we know that $f(X)$ is a compact subset of $\mathbb{R}^n$. Thus it is a bounded subset of $\mathbb{R}^n$ by the Heine-Borel theorem.
The Heine-Borel theorem mentioned above says:

A subset $E$ of $\mathbb{R}^n$ is compact iff $E$ is closed and bounded.

Thus compact subsets of $\mathbb{R}^n$ are bounded. However, the converse is not true. For instance, $(0,1)$ in $\mathbb{R}$ is bounded but not compact.
A: A continuous image of a connected set is connected.
Also a topological space $X$ is connected iff there does not exist a surjective continuous function from $X$ to $\{0,1\}$(or any set with two elements in the reals for instance).
Now for compactness-continuity  we have also these results:

If $f:X \rightarrow Y$ is a continuous bijection and $X$ ia compact topological space and $Y$ is a Hausdorf topological space then $f$ is a homeomorphism.

.

Every continuous function on a compact metric space $X$ is uniformly continuous on $X$

