Continuity and Differentiability

Let f(x)= casesx3+8; & x<0,\ x2-4; & x 0, cases and g(x)= cases(x-8)1/3; & x<0,\ (x+4)1/2; & x 0. cases Then the number of points, where the function g f is discontinuous, is .

The number of points, at which the function f(x)= \6x, 2+3x2 \+|x-1| |x2- 14 |, x (- , ), is not differentiable, is .

Let f(x)= casesex-1, & x<0\ x2-5x+6, & x 0 cases and g(x)=f(|x|)+|f(x)|. If the number of points where g is not continuous and is not differentiable are and respectively, then + is equal to .

The number of points in the interval [2, 4], at which the function f(x)= [x2-x- 12 ], where [ ] denotes the greatest integer function, is discontinuous, is .

The number of points of discontinuity of the function f(x) = [ x22 ] - [ x], x [0, 4], where [ ] denotes the greatest integer function, is .

If the function f(x) = ( x) - ( x) x - x is continuous at x = 0, then f(0) is equal to .

Let m and n be the number of points at which the function f(x) = \x, x3, x5, , x21 \, x R, is not differentiable and not continuous, respectively. Then m + n is equal to .

Let f(x) = cases 3x, & x < 0 \ \1 + x + [x], x + 2[x]\, & 0 x 2 \ 5, & x > 2 cases where [.] denotes greatest integer function. If and are the number of points, where f is not continuous and is not differentiable, respectively, then + equals .