I am learning Calculus. I have read a proof from MIT OpenCourseware on how differentiability implies continuity. Here's the link (https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/1.-differentiation/part-a-definition-and-basic-rules/session-5-discontinuity/MIT18_01SCF10_Ses5e.pdf).
I get the proof. Or rather, I understand why all the steps of the proof are true.
But I don't understand why all these steps prove that a function that is differentiable is continuous. It seems to me the author just took the derivative equation/limit, then rearranged it into the continuity definition.
Can someone explain to me, how being able to rearrange the derivative to form the continuity definition proves that differentiability = continuity? Can this be done without epsilon delta proofs and any knowledge of proof theory? Thank you!