Question

Let f : R → R be a function with the following property: f(x + y) = f(x) +f(y) for all x, y E R. 1. (1 point) Show that f(0) = 0. 2. (1 point) Show that f(-x) = -f(x). 3. (1 point) Show that f(x – y) =f(x) – f(y). 4. (3 points) Show: If n E N, then f (nx) = nf (x) and f(x/n) = (1/n)f(x) for all x. 5. (3 points) Show: If r E Q, then f (rx) = rf(x) for all x. 6. (3 points) Show: If f is continuous at xo=0, then f is continuous. 7. (3 points) Show: If f is continuous, then there exists c E R such that f(x) = cx. (Hint.c = f(1).)

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