# Find an example of a continuous function and open (closed) A such that f(A) is not open(closed)

for a continuous function and open A such that f(A) is not open. I think $f:R \rightarrow R$, and $f(A)=0$ for $A \in (0,1)$ works. since a singe point in $R$ is closed.

I have no idea to find a continuous function and closed A such that f(A) is not closed.

• just an example – user378456 Oct 25 '16 at 3:51
• If you are restricting yourself to functions $f:R\to R$, then $A$ must be a non-compact closed set. (Think about why this must be so.) – Braindead Oct 25 '16 at 3:55
• Kind of confusing that the question ends with (closed), since [closed] has meaning on SE – QuantumFool Oct 25 '16 at 4:12

## 1 Answer

Let $f(x) = e^{-x^2}$ and $A= \mathbb{R}$.

Then $f(\mathbb{R}) = (0,1]$ which is neither open nor closed. $A$ is both open and closed.

• why you put the -ve sign in the power of e? – user426277 Jun 3 '17 at 15:05
• Now thanks I understood why. – user426277 Jun 3 '17 at 15:10