Determine whether the following statements are true and give an explanation or counterexample. a. If the function f is differentiable for all values of x, then f is continuous for all values of x. b. The function f(x) = x- 1 is continuous for all x, but not differentiable for all x. c. It is possible for the domain of f to be (a,b) and the domain of f' to be [a,b]. a. Choose the correct answer choice below. O A. False, because there may be points at which f is differentiable, but not continuous. В. False, because 1 is differentiable everywhere, but it is not continuous at x = 0. Oc. True, because 1 is both differentiable and continuous everywhere. O D. True, because if f is differentiable everywhere, then that means there are no holes anywhere, and is therefore continuous everywhere. b. Choose the correct answer choice below. A. False, because f(x) is actually not continuous for all x, but it is differentiable for all x. O B. True, because f(x) is continuous, but not differentiable at the corner of this function which is at the point (1,0). O C. True, because f(x) has a hole in the graph at x = - 1; thus, it is not differentiable everywhere. D. False, because it is possible to draw a tangent line at every point on f(x). c. Choose the correct answer choice below. A. False, because by definition, f' cannot share any points in common with the domain of f. B. True, because it is still possible to draw a tangent line at a point that is not in the domain. C. False, because it is impossible to draw a tangent line at a point that is not in the domain. O D. True, because the endpoints of the domain of f' always contain an extra pair of points designated for tangent lines. O O O O

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.