218 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION Remark 3.1.2 can also be used to get a weak analogue of the Intermediate Value Theorem ( Calculus Deconstructed , Theorem 3.2.1). Recall that this says, for f : R → R continuous on [ a,b ], that if f ( a ) = A and f ( b ) = B then for every C between A and B the equation f ( x ) = C has at least one solution between a and b . Since the notion of a point in the plane or in space being “between” two others doesn’t really make sense, there isn’t really a direct analogue of the Intermediate Value Theorem, either for −→ f : R → R 3 or for f : R 3 → R . However, we can do the following: Given two points −→ a , −→ b ∈ R 3 , we deFne a path from −→ a to −→ b to be the image of any locally one-to-one continuous function −→ p : R → R 3 , parametrized so that −→ p ( a ) = −→ a and −→ p ( b ) = −→ b . Then we can talk about points “between” −→ a and −→ b along this curve . Proposition 3.1.4.
This is the end of the preview.
access the rest of the document.