4 - Continuity in metric spaces

This video continues the series, discussing continuity, first for functions on the real line, then in general metric spaces. It follows the top half of the eighth page of my course handouts, corresponding to the main concept in Chapter 4 in Carothers.
This video moves quickly (and has a lot of content)! Please rewind, rewatch, and/or slow the playback speed as necessary---once the whole thing makes sense in real time, you're in pretty good shape with the basic concepts!
Here's the link to the "Functions and Graphs" video: • Functions and Graphs

[@Deprisol]

My dude just made a espectacular, intuitive yet rigorous demonstration of Continuity and still hasn’t hit 400 views

[@frankswenton3177]

Thanks! No idea how to get into the algorithms. :)

[@jeonghoonchoi8175]

Hello, really appreciate these videos. Was just wondering if the next vid on properties of metric spaces ever published?

[@frankswenton3177]

Oh...this all started during COVID until I couldn't keep up with the class schedule, and I sort of forgot about it since then. Maybe I should put it on my to-do list! 😊

[@jeonghoonchoi8175]

⁠@@frankswenton3177 thank you 🙏

[@googaDEV]

Why the distortion of the ball in the Domain?

[@frankswenton3177]

If I'm understanding your question properly, the issue is that we have a nice round epsilon-ball in the codomain and another delta-ball in the domain, but the function definitely need not exactly map a ball to a ball. In general, a mapping (continuous function) could distort shapes in all sorts of ways, meaning that what starts out looking like a regular delta-ball in the domain could map to a quite different shape in the codomain. In the same way, the preimage of a regular ball in the codomain could be distorted as well. The notion of continuity can be expressed in terms of open balls, but it's always about something being either contained in or containing an open ball---those images are preimages relate to open balls, but they themselves need not _be_ open balls. I hope that helps a bit!