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4. (15 points) Show that is a differentiable function on R. Is x =0 the derivative f':R→R a continuous function?

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2.語言教學原則強調「結構、功能、文化」之結合。請以「道歉」為主題，運用上述原則於初級成人學生之華語口語能力的教學課程設計上，並說明你的教學重點為何？（15 分）

1. (10 points) Is =R? State your reason.

2. (10 points) Show that = 1.

3. (15 points) Is uniformly continuous on R? Prove your assertion.

5. (20 points) Show thatdx converges conditionally.

6. (10 points) Let fn:[0,1] →R, , be a sequence of increasing functions, i.e., fn(x) ≤fn(y) for all and n N. Assume that fn Sfnt1 and Ifn(x)I s 1 for all xe [0,1] and neN. Show that fn converges (pointwisely) to an increasing function.

7. (10 points) Can you find a C1 function f:R2 →IR such that Vf(x,y)= (-y,x) for all (x,y)? Find such a function or prove that it does not exist.

8. (10 points) Construct a function f:R2 →IR such that fx and fy exist at (0,0) but f is not differentiable at (0,0).

1. (10 %) Prove or disprove that.

2. (5%) Letdt, find derivative of F(x) .

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