The following are true or false variances of determining whether a function is continuous.

1. If $\displaystyle f:\mathbb{R} \rightarrow \mathbb{R} $ f(K) is bounded, then f is continuous.

False. A bounded step function such as the characteristic function would serve as a counterexample.

2. If $\displaystyle f:\mathbb{R} \rightarrow \mathbb{R} $ f(K) is sequentially compact whenever K is sequentially, then f is continuous.

False. The characteristic function serves as a counterexample

$\displaystyle f(x)=

\begin{cases}

1 & \text{if $x \in K$}, \\

0 & \text{if $x \notin K$}.

\end{cases} $ since it is bounded and closed, but it is not continuous.

(I know the converse is true: if f is continuous and K is sequentially compact, then f(K) is sequentially compact. )

3. If $\displaystyle f:\mathbb{R} \rightarrow \mathbb{R} $ f(A) is closed whenever A is closed, then f is continuous.

False. Consider the function

$\displaystyle f(x)=

\begin{cases}

tanx & \text{if $ - \frac{\pi}{2} < x < \frac{\pi}{2}$}, \\

0 & \text{if $otherwise$}.

\end{cases} $

It is closed on the real line but it is not continuous. Is there a better counterexample for this?

4. If $\displaystyle f:\mathbb{R} \rightarrow \mathbb{R} $ f(A) is open whenever A is open, then f is continuous.

False. I was thinking f(x) = 1/x. The range is $\displaystyle (- \infty, 0) \cup (0, \infty) $, which is open, but f is not continuous.

Thank you.