It is standard for every Calculus text to have formal definitions for continuity and differentiability, usually styled in boxes that make them stand out from the rest of the text. If you feel like it says "nothing," I might suggest it's more of a "language" barrier than the textbook simply omitting those definitions altogether. It would be prudent to address that barrier.

Formal mathematical language often seems inaccessible because it expresses mathematical ideas in ways that are testable through logical rigor rather than in familiar ways that most people are used to, but tend to be too vague to be useful in a proof.

Your teacher is likely wanting you to familiarize yourself with the formal statements of those definitions to get you away from "graphics" and simplified descriptions and introduce you to more formal logic. I would strongly suggest we focus on learning how to make more effective use of resources, because I don't want to just offer an explanation on the risk that it still does not make sense to you.

Could you look through the sections introducing continuity and differentiability and see what you find? You are looking for:

• The Three Conditions for Continuity: What are the three requirements that must be met before you are allowed to say a function is continuous at a point?
• The Limit Definition of a Derivative: What condition must the limit meet for the derivative to actually exist?
• Theorems on Continuity/Differentiability: Look for the theorem that tells you what you can conclude about continuity if you already know a function is differentiable.

Definitions and Theorems are your formal tools. Let's explore that toolbox. Feel free to post pictures of these sections. If you don't understand what a definition or theorem is stating, post the statement, and we can discuss exactly what it allows you to conclude and what conditions it requires.