Continuous and Discontinuous Functions

For what values of a and b is the function

f(x) `{:(= "a"x + 2"b" + 18",",  "for"  x ≤ 0),(= x^2 + 3"a" - "b"",",  "for"  0 < x ≤ 2),(= 8x - 2",",  "for"  x > 2):}}` continuous for every x?

Show that following function have continuous extension to the point where f(x) is not defined. Also find the extension :

f(x) = `(3sin^2 x + 2cos x(1 - cos 2x))/(2(1 - cos^2x)`, for x ≠ 0

Discuss the continuity of the following function at the point indicated against them :

f(x) = `{:(=( sqrt(3) - tanx)/(pi - 3x)",", x ≠ pi/3),(= 3/4",", x = pi/3):}}  "at"  x = pi/3`

Show that following function have continuous extension to the point where f(x) is not defined. Also find the extension :

f(x) = `(1 - cos2x)/sinx`, for x ≠ 0

If f(x) = `(4^(x - π) + 4^(π - x) - 2)/(x - π)^2` for x ≠ π, is continuous at x = π, then find f(π).

The following function has a removable discontinuity? If it has a removable discontinuity, redefine the function so that it becomes continuous :

f(x) `{:(=("e"^(5sinx) - "e"^(2x))/(5tanx - 3x)",",   "for"  x ≠ 0),(= 3/4",",   "for"  x = 0):}}` at x = 0

Discuss the continuity of the following function at the point(s) or on the interval indicated against them:

f(x) `{:(= (x^2 - 3x - 10)/(x - 5)",", "for"  3 ≤ x ≤ 6","  x ≠ 5),(= 10",", "for"  x = 5),(=(x^2 - 3x - 10)/(x - 5)",", "for"  6 < x ≤ 9):}`

Identify discontinuities for the following function as either a jump or a removable discontinuity :

f(x) `{:(= 4 + sin x",",  "for"  x < pi),(= 3 - cos x",",  "for"  x > pi):}`