Continuous and Discontinuous Functions

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]):}`

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

f(x) = `(cos4x - cos9x)/(1 - cosx)`, for x ≠ 0

f(0) = `68/15`, at x = 0 on `- pi/2 ≤ x ≤ pi/2`

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

If f(x) = `{{:((sin^3(sqrt(3)).log(1  +  3x))/((tan^-1 sqrt(x))^2(e^(5sqrt(3))  -  1)x)",", x ≠ 0),(                         a",", x = 0):}`

is continuous in [0, 1] then a is equal to ______.

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

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

f(x) `{:( = (sin^2pix)/(3(1 - x)^2) ",", "for"  x ≠ 1),(= (pi^2sin^2((pix)/2))/(3 + 4cos^2 ((pix)/2)) ",", "for"  x = 1):}}` at x = 1

Select the correct answer from the given alternatives:

f(x) `{:(= ((16^x - 1)(9^x - 1))/((27^x - 1)(32^x - 1))",", "for"  x ≠ 0),(= "k"",", "for"  x = 0):}` is continuous at x = 0, then ‘k’ =

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

f(x)  `{:(=(4^x - 2^(x + 1) + 1)/(1 - cos 2x)",",  "for"  x ≠ 0),(= (log 2)^2/2",",  "for"  x = 0):}}` at x = 0.