Timothy Biehler and Sean Maley propose teaching continuity and limits in a way that aligns with students' intuitions rather than the traditional epsilon-delta definition. They advocate using real-world examples like insulin dosing that demonstrate how continuity allows for small errors in measurement. This "fuzzy" approach explains limits and continuity as quantities being "around" or "close to" rather than exactly equal, mirroring how these concepts are applied in reality. They believe this intuitive understanding prepares students for later concepts like differentiability while increasing accessibility and relevance.

1. A “Fuzzy-Minded”
Approach to Continuity
Timothy Biehler, Professor of Mathematics, FLCC
Sean Maley, Assistant Professor of Mathematics, FLCC

2. Objective
• Teach continuity and limits in an introductory Calculus course in a way that
equips students with a clear sense of both the meaning and utility of these
concepts.

3. Traditional Approach: The “Epsilon-Delta”
Definition and Continuity

4. Modern, Intuitive Approach to
Limit and Continuity:
• Limits:
• Continuity:

5. Warm up

6. Warm up
(7.5 is the right answer. 7.48 is wrong. 7.56 is wrong.
Anything other than 7.5 is a wrong answer!)

7. Next problem
Tim has diabetes. The amount of insulin he needs to take before breakfast is a
function of the amount of carbs his breakfast will contain (in grams). He needs
2.5 units regardless of what he eats, and then one additional unit for each six
grams of carbs. How much insulin does he need to take if his breakfast will
contain 30 grams of carbs?

8. Look familiar?

9. In the math classroom
• We tend to speak in “exact”
terms
• In math class, f(30)=7.5, and 30
grams of carbs require 7.5 units
of insulin.
• Is it “exactly” 7.5 units?
Or “pretty much” 7.5 units?

10. • Here’s what an insulin syringe looks like:
• Think you can hit exactly 7.5 units?

11. Exact VS Reality
Google says the time required to get here from Canandaigua is 4 hours and 10 minutes.
Of course no one takes that to mean the exact time it took us to get from exactly FLCC to exactly here
is exactly 4hrs and 10 minutes!

12. What does this have to do with continuity?
• In reality, Tim didn’t eat exactly 30g of carbs and take exactly 7.5 units of
insulin.
• We understand the dose for around 30g of carbs is close to 7.5 units.
• Limits and continuity are what makes this OK.

13. In math class, we say f(30)=7.5 as though the values must be exact.

14. But we never say that.
But we really understand this as “f(30-ish)=7.5 or so”

15. What about limits?
Because when we are around
x=30, f(x) is around 7.5

16. Discontinuity
• On AT&T International Data plan, a package covering up to 120
MB of cellular data costs $30, and overages cost $0.25 per MB. A
package covering 300 MB of data costs $60, and you are switched
to that plan automatically once your usage reaches 300 MB.
• What is the cost of around 300 MB?
• 299MB?
• 301MB?

17. Discontinuity
• What is the cost of around 300 MB?

18. Advantages to the “Fuzzy Approach”
• Appeals to students’ common sense.
• Leads to more sound graphical understanding as well.
• Another way to say “small changes in input  small changes in
output”
Approachable
• Easier to explain why continuity is needed for differentiability
later onConnecting Ideas
• Clearly articulates the nature of continuity in realistic
applications.
(Rarely do students draw this connection on their own.)
Relevant

19. Making Epsilon and Delta Meaningful
• For any error tolerance in the output (epsilon) we can find a degree of
precision in the input (delta).
• If we can measure the dose within +/- units on the syringe, how close are we
on the carbohydrate target?

20. More to come…
• Thanks for coming! SLIDES: tiny.cc/fuzzycontinuity
• Tim Biehler, Professor of Mathematics, FLCC
• Timothy.Biehler@flcc.edu
• Sean Maley, Assistant Professor of Mathematics, FLCC
• Sean.Maley@flcc.edu