Hi Scholars!

This week we consider a standard result about continuous functions:

Lemma: If two functionsfandgare continuous and the image offis a subset of the domain ofg, then the compositiong∘fis continuous too.

To my great surprise, when talking with a student earlier this week, I realized I could not find a satisfactory diagram that illustrates an ε-δ argument for this result. There are, however, many proof write-up examples online. So, I was motivated to create this graphic.

The aim was to craft a diagram that this student could watch repeatedly to build intuition. In my experience, students understand the validity of a proof write-up upon seeing it; but, they have expressed confusion or frustration at it being counterintuitive (*i.e.* they say they would have never come up with it on their own) to have this two-stage approach to get an inequality as opposed to, say, choosing δ to be the minimum of two terms. To break the proof down, I use the following outline in this graphic.

*Stage 1: *Visualize each of the given terms.

*Stage 2: *Show the key inequality we aim to verify in the proof.

*Stage 3: *Show the continuity of *g* enables us to assert the existence of τ.

*Stage 4: *Show the continuity of *f* enables us to assert the existence of δ.

*Stage 5: *Iteratively apply the results in Stages 3 and 4 to get the key inequality.

Color coding was included to help students directly connect the mathematical expressions with their graphical representation. Although it would be more representative of metrics “in the wild” to use oval shaped for the δ-ball, τ-ball, or ε-ball, I left these as circles to avoid an distraction from the key argument in the proof. For sake of clarity, some simplification was also needed to keep the diagram from being overly crowded. For example, it would have been nice to show *f* applied to the δ-ball gives a proper subset of the τ-ball, and similarly for *g *applied to the τ-ball. It would be great to make an even simpler variation of this result in 1D for more introductory students. If you have any suggestions for improvements of this diagram or napkin sketch ideas for what it might look like in 1D, please let me know!

Stay Awesome.

Howard

p.s. Here is a YouTube video I made about this result.

Cool!😎