# Differentiability Implies Continuity

An important theorem concerning derivatives is this: If a function f is differentiable at x = a, then f is continuous at x = a. The proof begins with the identity that for all And therefore, Since both sides are finite, the function is continuous at x = a. The converse of this theorem is…

# 2019 CED Unit 1 – Limits and Continuity

This is the first of a series of blog posts that I plan to write over the next few months, staying a little ahead of where you are so you can use anything you find useful in your planning. Look for this series every 2 – 4 weeks. Unit 1 contains topics on Limits and…

# How to Tell your Asymptote from a Hole in the Graph.

The fifth in the Graphing Calculator / Technology series (The MPAC discussion will continue next week) Seeing discontinuities on a graphing calculator is possible; but you need to know how a calculator graphs to do it. Here’s the story: The number you choose for XMIN becomes the x-coordinate of the (center of) the pixels in the…

# Continuity

The definition of continuity of a function used in most first-year calculus textbooks reads something like this: A function f is continuous at x = a if, and only if, (1) f(a) exists (the value is a finite number), (2)  exists (the limit is a finite number), and (3)  (the limit equals the value). A…

# Continuity

Limits logically come before continuity since the definition of continuity requires using limits. But practically and historically, continuity comes first. The concept of a limit is used to explain the various kinds of discontinuities and asymptotes. Start by studying discontinuities. Types of discontinuities to consider: removable (a gap or hole in the graph), jump, infinite…