Which of the following is not continuous for all x

Which of the following functions is continuous for every value of x except x=0?

(a) For which x-values does the function appear to be not continuous? (Select all that apply.)012345(b) For which x-values does the function appear to be not differentiable? (Select all that apply.)012345

or each of the following functions R → R, say if it is continuous in the cofinite topology.You do not need to justify your answer.(a) f (x) = x2 + 2x + 5 [2,U]It is continuous because |f −1(a)| ≤ 2 for all a ∈ R.(b) g(x) = sin(x + 2) [2,U]It is not continuous because |g−1(0)| = ∞.(c) h(x) = e3x [2,U]It is continuous because |h−1(a)| ≤ 1 for all a ∈ R

If g(x) is continuous for all real numbers and g(3) = -1, g(4) = 2, which of the following are necessarily true?I. g(x) = 1 at least onceII. lim⁡𝑥→3.5𝑔(𝑥)=𝑔(3.5)III. lim⁡𝑥→3−𝑔(𝑥)=lim⁡𝑥→3+𝑔(𝑥)​ I. g(x) = 1 at least onceII.  x→3.5lim​ g(x)=g(3.5)III.  x→3−lim​ g(x)= x→3+lim​ g(x)​

Find each x-value at which f is discontinuous and for each x-value, determine whether f is continuous from the right, or from the left, or neither.f(x) =  x + 6      if x < 0 ex      if 0 ≤ x ≤ 1 6 − x      if x > 1x = (smaller value)continuous from the rightcontinuous from the left    neitherx = (larger value)continuous from the rightcontinuous from the left    neither