Final answer:
The set of continuous real valued functions on an interval is falsely claimed to be a subspace of R; in fact, it's a subspace of the vector space of all real-valued functions over that interval. Additionally, while y(x) must be continuous, the continuity of its first derivative is context-dependent and might not be required if a potential function is infinite.
Step-by-step explanation:
The statement "The set of continuous real valued functions on the interval is a subspace of R" is false. A subspace is a subset of a vector space that is itself a vector space under the same operations. For a subspace to exist, it must contain the zero vector, be closed under vector addition, and be closed under scalar multiplication. The set of all continuous real-valued functions over an interval satisfies these conditions, thus it is a subspace of the vector space of all real-valued functions on that interval, not of R itself, which is the set of real numbers.
As for the statements provided:
- y (x) must be a continuous function is true.
- The statement that "The first derivative of y(x) with respect to space, dy (x)/dx, must be continuous, unless V (x) = ∞" can be false, depending on the context. In classical mechanics, this statement could be a condition for a system described by a potential V (x). If V (x) is infinite, the system might involve a discontinuity in the derivative; however, outside of this context, the continuity of dy (x)/dx is not conditional on the value of a potential function.