Let be a function that satisfies the following properties: f(x+y)= f(x)f(y) for all r. y R,...

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Let be a function that satisfies the following properties: f(x+y)= f(x)f(y) for all r. y R, and f'(0) = 1. a. Explain why f is continuous at x=0. b. Show the following two facts: that f(0) = 1, and that f(x) 0 for all R. (You may prove them in either order, and use whichever you prove first to prove the second if you wish.) c. Show that f is continuous for all z R. d. Show that f is differentiable for all r ER, and compute f'(r) in terms of r and f(r). e. In fact, f is uniquely defined by the above properties, and is a well-known function. Which is it? Let be a function that satisfies the following properties: f(x+y)= f(x)f(y) for all r. y R, and f'(0) = 1. a. Explain why f is continuous at x=0. b. Show the following two facts: that f(0) = 1, and that f(x) 0 for all R. (You may prove them in either order, and use whichever you prove first to prove the second if you wish.) c. Show that f is continuous for all z R. d. Show that f is differentiable for all r ER, and compute f'(r) in terms of r and f(r). e. In fact, f is uniquely defined by the above properties, and is a well-known function. Which is it?