Continuous and Discontinuous Functions

Let f(x) = ax + b (where a and b are unknown)

= x² + 5 for x ∈ R

Find the values of a and b, so that f(x) is continuous at x = 1

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

f(x) `{:(= 2x^2 + x + 1",", "for"  |x - 3| ≥ 2),(= x^2 + 3",", "for"  1 < x < 5):}`

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

f(x) `{:(= (x^3 - 8)/(x^2 - 4)",",  "for"  x > 2),(= 3",",  "for"  x = 2),(= ("e"^(3(x - 2)^2 - 1))/(2(x - 2)^2) ",",  "for"  x < 2):}`

Identify discontinuity for the following function as either a jump or a removable discontinuity on their respective domain:

f(x) `{:(= x^2 + x - 3,","  "for"  x ∈ [ -5, -2)),(= x^2 - 5,","  "for"  x ∈ (-2, 5]):}`

If f(x) `{:(= (24^x - 8^x - 3^x + 1)/(12^x - 4^x - 3^x + 1)",",  "for"  x ≠ 0), (= "k"",",  "for"  x = 0):}}` is continuous at x = 0, find k

Test the continuity of the following function at the point or interval indicated against them:

f(x) `{:( =(x^2 + 8x - 20)/(2x^2 - 9x + 10)",",  "for"  0 < x < 3","  x ≠ 2),(= 12",",  "for"  x = 2),(= (2 - 2x - x^2)/(x - 4)",",  "for"  3 ≤ x < 4):}}` at x = 2

Examine whether the function is continuous at the points indicated against them:
f(x) = x³ − 2x + 1,         for x ≤ 2
      = 3x − 2,                 for x > 2, at x = 2

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

f(x) = `(x^2 - 1)/(x^3 + 1)` for x ≠ – 1