Let B be the closed ball in R2 with centre at the origin and radius unity. Pick out the true statements

Let B be the closed ball in $\Bbb R^2$ with centre at the origin and radius unity. Pick out the true statements.

(a) There exists a continuous function $f : B \to\Bbb R$ which is one-one.
(b) There exists a continuous function $f : B \to\Bbb R$ which is onto.
(c) There exists a continuous function $f : B \to\Bbb R$ which is one-one and onto.

continuous image of a connected (compact) space is connected (compact). but how can i use this result in this problem

I assume here that by R you mean $$\mathbb{R}$$ the set of real numbers. First notice that $$\mathbb{R}$$ is unbounded and thus not compact. You know that $$B$$ is compact. This is enough to handle the onto statement.
Let us consider the restriction of $$f$$ to the unit circle $$S$$. Since $$f(S)$$ must be compact and connected, it must be of the form $$[a,b] \in \mathbb{R}$$. If $$f$$ is 1-to-1 then $$a \not= b$$. Next, observe that if we remove any one of the uncountably infinite number of points from $$S$$ then we get a space that is homeomorphic to $$\mathbb{R}$$ and thus connected. On the other hand, there are only two points that we can remove from $$[a,b]$$ that will result in a connected space, the two points being $$a$$ and $$b$$. So, for any point $$x \in (a,b)$$, there are two distinct points in $$S$$ which gets mapped to $$x$$ . This should be enough to handle the 1-to-1 statement.