Definition of Continuity - Continuity of a Function at a Point

The value of p and q for which the function

f(x) = `{((sin ("p" + 1)x + sin x)/x","  x < 0),(q ","  x = 0),((sqrt(x + x^2) - sqrtx)/(x^(3//2))","  x > 0):}`

is continuous for all x in R, are

If f(x) = `{{:(4x - 2",", 0 ≤ x ≤ 2),(3x + "k"",", 2 < x ≤ 3):}` is continuous in [0, 3], then the value of k is ______.

If f(x) = x² + α for x > 0 = `2sqrt(x^2 + 1)` + β for x < 0 is continuous at x = 0 and `f(1/2)` = 2 then α² + β² is ______.

If f (x) = x 2 + a, for x ≥ 0 ` =2sqrt(x^2+1)+b, ` is continuous at x = 0, find a and b.

Find a and b, so that the function f(x) defined by

f(x)=-2sin x,       for -π≤ x ≤ -π/2

     =a sin x+b,  for -π/2≤ x ≤ π/2

     =cos x,        for π/2≤ x ≤ π

is continuous on [- π, π]

lf f(x) = `(20^x + 3^x - 6^x - 10^x)/(1 - cos8x)`, for x ≠ 0

= `(k/16)log(10/3).log2`, for x = 0

is continuous at x = 0, then the value of k is ______ 

if the function

`f(x)=k+x, for x<1`

       `=4x+3, for x>=1`

id continuous at x=1 then k=

(a) 7

(b) 8

(c) 6

(d) -6

if  `f(x) = (x^2-9)/(x-3) + alpha`               for x> 3

           =5,                                     for x = 3

          `=2x^2+3x+beta`,             for x < 3

is continuous at x  = 3, find α and β.