**Context: **This lesson plan is from a weekly communication recitation that accompanies M.I.T.’s *Real Analysis* (18.100C). This week students learn about differentiability and the mean value theorem (Rudin pp. 103-110). This recitation revisits concepts that were introduced in the second recitation.

**Authors: **The recitation was developed primarily by Joel B. Lewis and Craig Desjardins.

**Communication objectives:** Translating among mathematical concepts, mathematical language, and notation; with particular attention to how changing the order of quantifiers changes the meaning.

Students worked in small groups on the following task:

Give some examples of f: **R**–>**R** having each of the following properties.

- A function f: X–>Y is said to have property P
_{1}if for all ε > 0 and for all x_{1}, x_{2}in X there exists δ > 0 such that if d_{1}(x_{1}, x_{2}) < δ, we have that d_{2}(f(x_{1}), f(x_{2})) < ε. - A function f: X–>Y is said to have property P
_{2}if for all ε > 0 there exists δ > 0 such that for all x_{1}, x_{2}in X with d_{1}(x_{1}, x_{2}) < δ, we have that d_{2}(f(x_{1}), f(x_{2})) < ε. - A function f: X–>Y is said to have property P
_{3}if there exists δ > 0 such that for all ε > 0 and for all x_{1}, x_{2}in X with d_{1}(x_{1}, x_{2}) < δ, we have that d_{2}(f(x_{1}), f(x_{2})) < ε.

Then, as class, we discussed the meaning of each of the statements above. The following points came up in discussion:

- Note that f is continuous on its domain if for all x
_{1}in X and for all ε > 0 there exists δ > 0 such that for all x_{2}in X with d_{1}(x_{1}, x_{2}) < δ, we have that d_{2}(f(x_{1}), f(x_{2})) < ε. [This is not one of the listed properties.] - Note that P
_{2}is uniform continuity, which can be illustrated by drawing windows on a graph. - P
_{1}is satisfied by every function. - On the reals, the only functions satisfying P
_{3}are constant functions, but this is not true in every metric space.

**Note:** The above activity does not fill an entire recitation. In Fall 2009 the activity was combined with the conclusion of peer critique that had been started in the preceding recitation.