To determine if set is open or closed Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f (t) = t^{2} $ and let $U$ be any nonempty open subset of $\mathbb{R}$ Then 
a .$ f (U) $ is open
b.  $f (U) $ is closed
c.  $f^{-1}(U)$ is open 
d. $f^{-1}(U)$ is closed
Attempt  is that I take any set $U$ consisting of elements say $1,2,3$ then $f (U)$ will have $1 ,4,9$ .Now limit point of $f (U)$ is $\phi $ and hence it is closed but answer seems to be $c$.
Can anyone help where I went wrong?
 A: Hint: If $f$ is continuous then $f^{-1}(U)$ is open to any $U \subset \mathbb{R}$ open set. 
Edit: Consider $f: M \to N$ continuous.
Let $A' \subset N$ be an open set, we want to show that $f^{-1}(A')$ is open. In fact, for each $a \in f^{-1}(A')$, we have that $f(a) \in A'$. By definition of open set, there exists $\epsilon > 0$ such that $B(f(a), \epsilon) \subset A'$. As $f$ is continuous at $a$, there exists $\delta > 0$ correspondent to $\epsilon > 0$ such that 
$$f(B(a; \delta)) \subset B(f(a); \epsilon) \subset A'$$
that is, $$B(a;\delta) \subset f^{-1}(A')$$
Then $f^{-1}(A')$ is open.   

A: Statement c is of course true, because the function is continuous.
If you know that the function is continuous (at every point), then take an open set $U$. In order to show that $f^{-1}(U)$ is open, we need to see that, for $x\in f^{-1}(U)$, there exists $\delta>0$ with 
$$
B(x,\delta)=\{r\in\mathbb{R}:|r-x|<\delta\}\subseteq f^{-1}(U).
$$
Since $f$ is continuous at $x$ and $U$ is open, there exists $\varepsilon>0$ such that
$$
B(f(x),\varepsilon)\subseteq U.
$$
Now apply the $\varepsilon$-$\delta$ definition of continuity and the problem is solved: the required $\delta>0$ is exactly the one provided by the continuity condition.
Why is the function continuous? By general theorems: the product of continuous functions is continuous; the identity function $t\mapsto t$ is obviously continuous.

The other statements are false, in the sense that for each one you can find a particular $U$ such that the condition doesn't hold.
For a consider $U=\mathbb{R}$. Hint: $f(U)=[0,\infty)$
For b consider $U=(0,\infty)$. Hint: $f(U)=(0,\infty)$
For d consider $U=(0,1)$. Hint: $f^{-1}(U)=(-1,0)\cup(0,1)$
