A question on limits, continuity and differentiability.

Question

1) Find the values of constants c and k that make f(x) continuous on real numbers.

                              {    x + 2c,     if x < -2,

                     f(x) = {   3cx + k,    if -2≤ x ≤ 1,

                              {    3x - 2k,     if x >1.

      * The bracket { is supposed to be a big one. So instead I put down three of them. :)

2) Let

                             { x^2 + 2x,        if x< -1  

                   g(x) = { 0,                   if -1 ≤ x ≤ 1,

                             { ln x,                if x>1.

    a) Where is g(x) not continuous ?                     

    b) Where is g(x) not differentiable?

Answer 1

1) x+2c=3cx+k when x=-2 for continuity. So -2+2c=-6c+k, 8c=k+2, k=8c-2.

Also, 3cx+k=3x-2k when x=1 for continuity. So 3c+k=3-2k, 3k=3-3c.

Substitute for k: 3(8c-2)=3-3c, 24c-6=3-3c, 27c=9, c=⅓.

Therefore 3k=3-1=2 and k=⅔.

2)

a) g(x) is discontinuous at x=-1, because when x<-1 and x→-1, g(x)→-1, but when x=-1, g(x)=0.

b) g(x) is not differentiable at x=1