Intermediate Mathematical Methods (Sequences and continuity)

Answer all parts of all questions
1. By constructing a suitable sequence {xn} to the transition point, discuss whether the function is (i) continuous (ii) C1 or (iii) Ck where k > 1. (3 X 10 marks)
a.    f(x) = x2 +1 if x <1 = 3 if x = 1 ?
b.    f(x) = x2 +1 if x <1 = 2 if x = 1 ?
c.    f(x) = x2 +1 if x <1 = 2x if x = 1 ?
2. Consider the function f(x,y) = 2x2y/(x4+y4). Show that f1(t) = f(t,a) and f2(t) = f(a,t) are continuous functions of t for each fixed a. Show that f itself is not continuous at (0,0). [Hint: Take a sequence on the diagonal] (20 marks)

Intermediate Mathematical Methods (Sequences and continuity)

Answer all parts of all questions
1. By constructing a suitable sequence {xn} to the transition point, discuss whether the function is (i) continuous (ii) C1 or (iii) Ck where k > 1. (3 X 10 marks)
a.    f(x) = x2 +1 if x <1 = 3 if x = 1 ?
b.    f(x) = x2 +1 if x <1 = 2 if x = 1 ?
c.    f(x) = x2 +1 if x <1 = 2x if x = 1 ?
2. Consider the function f(x,y) = 2x2y/(x4+y4). Show that f1(t) = f(t,a) and f2(t) = f(a,t) are continuous functions of t for each fixed a. Show that f itself is not continuous at (0,0). [Hint: Take a sequence on the diagonal] (20 marks)