Q.5 Let݂ $S=[0,1)\cup [2,3]$ and $f:S\to R$ be a strictly increasing function such that $f(S)$ is connected. Which of the following statements is TRUE?
(A) ݂$f$ has exactly one discontinuity.
(B)݂$f$ has exactly two discontinuities.
(C)݂$f$ has infinitely many discontinuities.
(D)݂$f$ is continuous.
Can someone give some hints how to tackle this problem.
Hints:
The connected subsets of $\mathbb{R}$ are intervals. (might be open, closed, half open, bounded, unbounded, it doesn't matter. They are intervals). How can you describe the image of $f$ then?
Because the function is increasing it has one sided limits at every point.
Try to combine both hints and it will get you to the solution.