If f(x) is discontinuous at x=c, then limx→c- f(x) ≠ limx→c+ f(x)
True or False?
If f(x) is discontinuous at x=c, then limx→c- f(x) ≠ limx→c+ f(x) True or False?
True or False (a) If X ∩ Y = ∅ then the two events X and Y are independent? (b) If event X is independent of event Y, then X^c is independent of Y? (c) For a discrete random variable X, we have limx->∞ pX(x) = 0? (d) For a continuous random variable X, we have limx->∞ fX(x) = 0? (e) For a continuous random variable X, we have limx->0 fX(x) ≤ 1? (f) For two discrete random variables X...
= (a) Suppose that limx+c f(x) L > 0. Prove that there exists a 8 >0 such that if 0 < \x – c < 8, then f(x) > 0. (b) Use Part (a) and the Heine-Borel Theorem to prove that if is continuous on [a, b] and f(x) > 0 for all x € [a,b], then there exists an e > 0 such that f(x) > e for all x E [a, b].
Question For this problem, consider the function y=f(x)= |x| + x 3 on the domain of all real numbers. (a) The value of limx→ ∞f(x) is . (If you need to use -∞ or ∞, enter -infinity or infinity.) (b) The value of limx→ −∞f(x) is . (If you need to use -∞ or ∞, enter -infinity or infinity.) (c) There are two x-intercepts; list these in increasing order: s= , t= . (d) The intercepts in part (c) divide...
1 6. Where is the function f(x) { { - X4 if x # 0 discontinuous? if x = 0 0 Is this a removable discontinuity? ex if x < 0 7. Where is the function f(x) discontinuous? x2 if x > 0 Is this a removable discontinuity? Is it a jump discontinuity? f(x) = {
ind for which values of x the follow ction is discontinuous: X – 3 f(x) = - x2 + 7x If there is more than one value, separate your answers by comas. The function is discontinuous at x =
If F(x)is an antiderivative of f (x), then f (x) = F(x). True False Previous
11. Circle true or false. No justification is needed. (14 points) (a) If f(x) - o(g(x), and both functions are continuous and positive, then fix dz converges. TRUE FALSE (b) If f(x)- o(g(x)), then f(x)gx)~g(x). TRUE FALSE (c) If the power series Σ an(x + 2)" converges atェ= 5, then it must km0 converge at =-6. TRUE FALSE (d) There exists a power series Σ akz" which converges to f(z)-I on some interval of positive length around FALSE TRUE (e)...
a) Show that f is discontinuous at any x 6= 0. b) Show that f is continuous at x = 0. c) Show that f is differentiable at x = 0 and compute the value f 0 (0). d) Show that f is not integrable on the interval [1, 2] (or any interval, but I don’t mind if you use that interval specifically). (x2 (x EQ) f(x)=o (x &Q)
Problem 25. Letf : [a, b] → R be an increasing function. Show that limx→a f(x) exists. What can you say about the relationship between this limit and f(a)? Problem 26. Letf,g: R → R be two continuous functions. Define h(x) = max {f(x), g(x)} for all x E R. Show that h is continuous on R.
6. Where are the following functions discontinuous? a) f(x) = x+2) x+2 (x+2)x b) f(x) = 21