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Consider a theorem of the form: (Something) iff $\exists$ a continuous function $f: X \rightarrow Y$ s.t. (something about the function).

I'm unsure how one would go about proving the $(\rightarrow)$ direction.

So we suppose that (Something) is true, and we WTS that there exists a continuous ...

But in terms of showing the part on the right, do we let $f$ be a continuous function and then show the s.t. portion? Or do we let $f$ be a function s.t. (something about the function) and then show that $f$ is continuous? Or am I only allowed to let $f: X \rightarrow Y$ and then have to show $f$ is continuous and the s.t. (something about the function) follows as well?

Thanks!

T. Fo
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  • To convert the theorem into "if p, then q", you need to make sure what you want to implicate: continuity of f or the s.t. part of a continuous function.
    If you want to prove the existence of continuous function, you can take s.t. portion into if clause, i.e., hypothesis part.     – spkakkar Jan 28 '18 at 20:41

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Do we let $f$ be a continuous function and then show the s.t. portion?

No, because then you would show that every continuous function has that portion, which is not always true.

Or do we let $f$ be a function s.t. (something about the function) and then show that $f$ is continuous?

No, because then you would show that every function with that portion is continuous, which is also not always true.

Or am I only allowed to let $f:X\to Y$ and then have to show $f$ is continuous and the s.t. (something about the function) follows as well?

No, because then you would prove that every function is continuous and has that portion. Which is not true.

What you need to do: assume (something), then construct a continuous function with that portion. So you really need to make a continuous function yourself and prove that it has that portion (all based on the assumption of something).