My advice is to pick one or the other, not both, for topic, probably continuity.
Also, physics students are not a-mathematical. They may not be PhD geometrists. But they are definitely people that cleave to the standard algebra and calculus track.
I would avoid the trap of thinking of them as unwashed non Rudinites that you are going to correct gaps of. Instead, frame your talk more to explore the different ways one can think about continuity, or perhaps with a msth history flair, discuss how and who had different frames over time. On other words more like a fun pop video on YouTube.
It is fine to mention some very advanced concepts without feeling you need to teach and prove. I get value out of knowing what a topic is, who works on it, what classes cover it, without knowing the content itself. Like having a situational awareness of what an abstract algebra course us, without having ever done one. Similar to how you (should) have a general awareness of what an organic chemistry course is, without having done one. But more as a generally educated person to know whatvthe fields of knowledge are.
Format is separate from the aspect of how you slant the talk. But my advice is to do a chalk talk. And even for that, I would do more of an entertaining speech, not a step by step derivation. Ie less chalk than a teacher. Just occasionally as needed, for some example or diagram.
Don't do slides. Will put them to sleep. Death by PowerPoint. And it will make it to easy for you to fall into the trap of being too hard and too proofy.