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If you have any doubt or need more clarification at any step please comment.
definition of continuity to prove that f : (0,00) by f(x)-13 + 1 is continuous at...
7. Using the definition of continuity directly to prove that f: (1,00) + R defined by f(3) = and f(1) = 0 is continuous on at 2 = 2 but not continuous at 1.
Use the definition of uniform continuity to prove that f(x)is uniformly continuous on , 00
PLEASE ANSWER ALL! SHOWS STEPS 2. (a) Prove by using the definition of convergence only, without using limit theo- (b) Prove by using the definition of continuity, or by using the є_ó property, that 3. Let f be a twice differentiable function defined on the closed interval [0, 1]. Suppose rems, that if (S) is a sequence converging to s, then lim, 10 2 f (x) is a continuous function on R r,s,t e [0,1] are defined so that r...
[4 Pts. Use the definition of continuity to show that the function f is continuous at <=0 10 g(x)= 3-4
Use the definition of continuity to determine whether f is continuous at a x2 - 169 f(x) = if x #13 4 if x = 13 a = 13 The function | at 13 is continuous is not continuous
2. Let f:R + R and g: R + R be functions both continuous at a point ceR. (a) Using the e-8 definition of continuity, prove that the function f g defined by (f.g)(x) = f(x) g(x) is continuous at c. (b) Using the characterization of continuity by sequences and related theorems, prove that the function fºg defined by (f.g)(x) = f(x) · g(x) is continuous at c. (Hint for (a): try to use the same trick we used to...
10. (a) 5pts] Prove that if f is piecewise continuous on [0,00) and satisfies the growth is, f(t) Me for restriction that some constants M k>0 on [0,00), then lim f(t)]=0 (b) [5pts] Find the inverse transform of s+1 F(6) s) = ln (c) [8pts] Find a solution of ty"+2y-ty= 0, y(0)=1, y (1)= Hint) You may need to use (a) and (b). 10. (a) 5pts] Prove that if f is piecewise continuous on [0,00) and satisfies the growth is,...
Format requirement: Question 3. E-6 Proof (Show Working) 10 points 249 Show that f:RR defined by f(x) is continuous at x = 7 using only r +3 cosa the epsilon-delta definition of continuity. Note that we want you to do it the hard way: you are not allowed to use the limit laws or the combination of continuous functions theorem or similar. You must give an 'e-δ style proof Solution: Let ε > 0 be given and choose δ =...
9. Prove that the function f(x) = ax+b is uniformly continuous on R by directly applying the e, 8 definition of uniform continuity.
PROBLEM 2: THE INDICATOR FUNCTION OF THE RATIONAL NUMBERS For a while, it was believed that any given function should be mostly continuous. This is reasonable, given the types of functions one typically sees in Calculus courses, where the worst case scenario involves a function that is defined piecewise and is continuous everywhere, except for some finite set of discontinuities, where the value of the function drops or jumps. It was also believed that every function should be integrable, which...