Contrapositive of theorem and issue with proof

Consider the following theorem:

Theorem Suppose that ##f:X\to Y## is a continuous map between two topological spaces ##X## and ##Y##. Then ##f(X)## is connected if ##X## is.

First, I don't know how to take the contrapositive of this statement. I'm not sure if the opening hypothesis, that is, ##f## is continuous, remains outside. Because the way this is proved in the text I'm reading is by assuming ##f(X)## is disconnected and then using continuity to show that ##X## is also disconnected. Since the author uses the continuity assumption, it seems like that is not a part of the statement that one takes the contrapositive of.

Yet, I saw someone claim,

Claim If a function has a connected domain ##X## and a disconnected range ##f(X)##, then the function is not continuous on ##X##.

So in this claim it seems like that the continuity is actually negated as well. I'm confused and I'm now doubting the validity of the proof of the theorem. What is correct and what isn't?

A: A function f(X) is a continuous map between two topological spaces X and f(X)
B: f(X) is connected
C: X is connected

If C and Not B, then Not A.

So the original sentence is When A: Then B, if C.
This can be rewritten: If A, then (if C then B)
The contrapositive: If not (if C then B) then not A.

(if C then B) is false only when C is true and B is false
so: not (if C then B) == ( C and not B)
Thus: If not (if C then B) then not A == if (C and not B) then not A

QED

psie said:

So in this claim it seems like that the continuity is actually negated as well. I'm confused and I'm now doubting the validity of the proof of the theorem. What is correct and what isn't?

The theorem itself is of the form:

Suppose ##f## is continuous, then property A holds.

The contrapositive of that is:

If property A does not hold, then ##f## is not continuous.

That's probably the answer for the contrapositive of the theorem.

In this case, property A is also an "if-then":

If ##X## is connected, then ##f(X)## is connected.

The contrapositive of property A is:

If ##f(X)## is not connected, then ##X## is not connected. But, that's not the contrapositive of the theorem.