Continuity on a discontinuous interval

Question

I got a question which asked me to prove the Intermediate Value Theorem for $f:X\to X$, where $X = [0,1] \cup [2,3]$ and $f$ is continuous.

I assumed that the function is either in $[0,1]$ or $[2,3]$, and so for the domain, but I am not sure if this is right.

Please guide me through this.

Answer 1

The Intermediate Value Theorem is not applicable in this case. For instance, the function $f:X\to X$ defined by $$f(x) = \begin{cases} 0, & \text{if $x\in[0,1]$} \\ 1, & \text{if $x\in[2,3]$} \end{cases} $$ is continuous on $X$, but clearly does not satisfy the Intermediate Value Theorem.

Answer 2

The fact that $f$ is continuous means that $f$ is continuous on the two separate intervals. There is no information on what values "between $f(1)$ and $f(2)$" the function reaches.

So, given that I understood your question correctly, "proving the Intermediate Value Theorem" on this domain is the same as proving it for the two separate intervals in the standard way.

The Intermediate Value Theorem is thus only applicable on each of the two separate intervals, but not on the whole domain.