Question

Prove that if f is continuous and f(x+y)=f(x)+f(y) for all x and y, then there is a number c such that f(x)=cx for all x. (This conclusion can be demonstrated simply by combining the results of two previous problems.) Point of information: There d0​ exist noncontinuous functions f satisfying f(x+y)=f(x)+f(y) for all x and y, but we cannot prove this now; in fact, this simple question involves ideas that are usually never mentioned in any undergraduate course. The Suggested Reading contains references.

Answer 1

If f is a continuous function such that f(x+y) = f(x) + f(y) for all x and y , then there exists a constant c such that f(x) = cx \) for all x .

Step-by-step explanation:

To prove this, let's consider the function g(x) = f(x) - cx . We want to show that g(x) = 0 for all x , which implies f(x) = cx .

1. Show that g(x) = 0 for some x :

`\[ g(0) = f(0) - c \cdot 0 = f(0) = 0 \]`

Since
`\( g(0) = 0 \)`, this implies that g(x) = 0 for some x by the Intermediate Value Theorem.

2. Show that g(x) = 0 for all x :

Consider
`\( h(x) = g(x)/x \). If \( g(x) \)` is not always zero, there exists
`\( x_1 \)`such that
`\( h(x_1) \\eq 0 \)`. By continuity of f and h , h(x) is not zero in a neighborhood around
`\( x_1 \).` However, this leads to a contradiction since h(x) = g(x)/x would then be undefined at x = 0 , violating the continuity of h .

3. Conclude
`\( f(x) = cx \) for all \( x \)`:

Since
`\( g(x) = 0 \)`for all x , we have
`\( f(x) - cx = 0 \),`implying f(x) = cx for all x . Therefore, the original function f can be expressed as
`\( f(x) = cx \)`, where c is a constant.