Consider the following function. f(x) sin χ ,x % 0 ,x=0 Prove that f is discontinuous at x=0 Prove that f is continuous for x = 0

2. PAST PAPER QUESTION (2018): Let f be a function that satisfies the following properties: f(x + y) = f(x)f(y) for all x, y ER, and ƒ'(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 x E 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 x € R. d. Show that f is differentiable for all x ER, and compute f'(x) in terms of x and f(x). e. In fact, f is uniquely defined by the above properties, and is a well-known function. Which is it?

(A) B (C) f (x) Let f be the function defined above. For what values of b is f continuous at x = 3? D = 0.394 only ebx 0.274 only for x3 x+b for x > 3 -4.500 and 0.394 -1.998 and 0.274

Can you please help me with this question?