Suppose $$f:[0,1]\times[0,1] \to (0,1)\times (0,1)$$ is a continuous,

non constant function. Which of the following statements 

are NOT true ?

Correct option is D

​$$\textbf{Option A and C:}$$​​

f is given continuous $$\implies$$​f maps compact set to compact set. 

​$$\implies \textbf{statement C true.}$$​​

Furthermore, range is given (0,1) $$\times$$​ (0,1) So , 

the image of $f$ must be uncountable, as a compact subset of a metric space like

$(0,1)\times (0,1)$ that is nontrivial must be uncountable.​

$$\textbf{statement A is true. }$$​​

​$$\textbf{Option B}$$​​

The function $f$ is continuous, and the domain $[0,1]\times [0,1]$ is path-connected.

The continuous image of a path-connected set is also path-connected.

Therefore, the image of $f$ is path-connected.​

​$$\textbf{Hence, statement given in option B is also true}$$

as all statements given in Option A,B and C are true. Option D must be correct.

Also , From given conditions f need not have empty interior.

​$$\textbf{So, statement given in option D is false }$$​​