The function that is 1 for x ≥ 0 and 0 for x < 0 is total and is not continuous.
I believe the intermediate value theorem does hold in the usual way it is stated.
I believe there are infinitely many infinite cardinalities.
I believe that CH is indeed a meaningful statement.
However, I do not believe it has a definite truth value. The truth value depends on what axioms we choose to use.
I believe that large cardinal axioms are meaningful statements.
However, I do not believe they have a definite truth value. Whether we choose to accept a large cardinal axiom as being true is a pragmatic choice. If it is interesting for set theorists to assume the existence of large cardinals because it is fruitful for research, then let them assume it to be true.
Build two machines, one with the "yes" light on and one with the "no" light on. Then you can be certain that you have built at least one machine that answers the question.
There are some mathematical questions that do not get determined by our choice of axioms. Thus, if you wanted to build one and only one machine, it would have no chance of determining the truth value of an arbitrary mathematical statement.
This depends on the semantics of "implies". If we are thinking about implication in the sense of classical propositional logic, then yes. If we have a notion of implication closer to the natural language notion of implication, then no.
Constructive proofs provide a lot of insight in terms of how to explicitly construct certain mathematical objects. However, this type of constructive insight comes at a price and it may sometimes be easier to follow and understand a classical non-constructive proof than a very detailed constructive proof of the same theorem.
For example, classical mathematics and constructive mathematics are different ways of doing mathematics with somewhat different rules about inference. They each constitute a different language game to be played.
There are mathematical truths that are more widely accepted than others. For example, "2 = 2" is more widely accepted than "there exists a large cardinal". However, this does not make the former more true than the latter.