In Exercises 1 through 28, determine whether each of the given sets is a real linear space, if addition and multiplication by real scalars are defined in the usual way. For those that are not, tell which axioms fail to hold. The functions in Exercises 1 through 17 are real-valued. In Exercises 3, 4, and 5, each function has domain containing 0 and 1. In Exercises 7 through 12, each domain contains all real numbers.
5. All $f$ with $f(1) = 1 + f(0)$.
Let $f$ and $g$ be elements in the set, with addition of functions defined as $(f + g)(x) = f(x) + g(x).$ We then have:
\begin{align*}
(f + g)(1) &= 1 + (f + g)(0)
\\
\\
&= 1 + f(0) + g(0)
\\
\\
&= f(1) + g(0)
\\
\\
&\neq f(1) + g(1)
\end{align*}
Hence, set of all $f$ with $f(1) = 1 + f(0)$ fails to satisfy the axiom under closure by addition and is not a linear space. $\,\blacksquare$
