understanding the halting problem

I was reading my Professor Richard Cole's note in the halting problem and I was confused on the statement for a while.

Understanding the following two examples will help a lot in understanding the Halting Problem.

Class Photographer, mutation of babar's paradox

In the class Note, yet figure the graph out.

A simpler and more abstract math model

Let \(H = \{ f: ran f = \{0, 1\} \}\), such \(H\) does not exist.

BWOC (By way of contradicton): Assume such \(H\) exists, then let \(h \in H\) s.t. \(h(f) = 1 - f \ \forall f \in H\) defined pointwise, then we have that \(h(h) = 1 - h\), then \[ h = 1 \iff h(h) = h = 0 \] Contradiction.