We study the function f defined below: 24 f(x) = if x is rational -x4 if...

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We study the function f defined below: 24 f(x) = if x is rational -x4 if x is irrational Prove that is nowhere continuous except at x=0. That is, you need to prove the following two statements. (1) (2 marks) f is continuous at x = 0. (2) (3 marks) f is not continuous at x = a for any a 0. Consider the function 9(2)-(2- 2-Tx if xsa if x > a Use the IVT (Intermediate Value Theorem) to justify that there exists a value of a which makes g continuous everywhere. 1. Let's recall the Pinching (Squeeze) theorem (Theorem 2.5.1): Suppose there are three functions f, g, and h that satisfy the following inequalities on a given interval containing a constant c. - h(x) f(x) g(x) for all x such that 0 < |x c| < p for some p > 0 If the functions h and g have the same limits as x approaches c, then the function f also has the same limit. That is, if lim h(x) = lim g(x) = L, for some real number L, then lim f(x) = L. xC xC xc Now, our goal is to generalize this theorem. The following is our initial incorrect attempt. You are going to fix it. 1 Suppose there are three functions f, g, and h that satisfy the following inequalities h(x) < f(x) < g(x) for all x such that 0 < |x c| < p for some p > 0 - and lim h(x) = H, lim f(x) = F, and limg(x) = G, for some real numbers H, F, and G. xC We claim that x-c xC H 9. Determine the following limit x-4 lim x 2 x 2 - We study the function f defined below: 24 f(x) = if x is rational -x4 if x is irrational Prove that is nowhere continuous except at x=0. That is, you need to prove the following two statements. (1) (2 marks) f is continuous at x = 0. (2) (3 marks) f is not continuous at x = a for any a 0. Consider the function 9(2)-(2- 2-Tx if xsa if x > a Use the IVT (Intermediate Value Theorem) to justify that there exists a value of a which makes g continuous everywhere. 1. Let's recall the Pinching (Squeeze) theorem (Theorem 2.5.1): Suppose there are three functions f, g, and h that satisfy the following inequalities on a given interval containing a constant c. - h(x) f(x) g(x) for all x such that 0 < |x c| < p for some p > 0 If the functions h and g have the same limits as x approaches c, then the function f also has the same limit. That is, if lim h(x) = lim g(x) = L, for some real number L, then lim f(x) = L. xC xC xc Now, our goal is to generalize this theorem. The following is our initial incorrect attempt. You are going to fix it. 1 Suppose there are three functions f, g, and h that satisfy the following inequalities h(x) < f(x) < g(x) for all x such that 0 < |x c| < p for some p > 0 - and lim h(x) = H, lim f(x) = F, and limg(x) = G, for some real numbers H, F, and G. xC We claim that x-c xC H 9. Determine the following limit x-4 lim x 2 x 2 -