If `f*g` is continuous at x=a, then f and g are separately continuous at x=a.

The function f(x)={(e^(1/x)-1)/(e^(1/x)+1),x!=0, at x=0 ,f(x)=0 a. is continuous at x=0 b. is not continuous at x=0 c. is not continuous at x=0, but can be made continuous at x=0 (d) none of these

Statement I f(x) = sin x + [x] is discontinuous at x = 0. Statement II If g(x) is continuous and f(x) is discontinuous, then g(x) + f(x) will necessarily be discontinuous at x = a.

If f(x) = 3(2x + 3)^(2//3) + 2x + 3 , then: (a) f(x) is continuous but not differentiable at x = - (3)/(2) (b) f(x) is differentiable at x = 0 (c) f(x) is continuous at x = 0 (d) f(x) is differentiable but not continuous at x = - (3)/(2)

Let f(x+y)=f(x)+f(y) for all xa n dydot If the function f(x) is continuous at x=0, show that f(x) is continuous for all xdot

A function f(x) satisfies the following property: f(x+y)=f(x)f(y)dot Show that the function is continuous for all values of x if its is continuous at x=1.

If f is continuous and g is a discontinuous function then f+g is continuous function.

if f(x) ={{:( (x log cos x)/( log( 1+x^(2) )), x ne 0) ,( 0, x=0):} a. f is continuous at x = 0 b. f is continuous at x = 0 but not differentiable at x = 0 c. f is differentiable at x = 0 d. f is not continuous at x = 0

Let y = f(x) be defined in [a, b], then (i) Test of continuity at x = c, a lt c lt b (ii) Test of continuity at x = a (iii) Test of continuity at x = b Case I Test of continuity at x = c, a lt c lt b If y = f(x) be defined at x = c and its value f(c) be equal to limit of f(x) as x rarr c i.e. f(c) = lim_(x to c) f(x) or lim_(x to c^(-))f(x) = f(c) = lim_(x to c^(+)) f(x) or LHL = f(c) = RHL then, y = f(x) is continuous at x = c. Case II Test of continuity at x = a If RHL = f(a) Then, f(x) is said to be continuous at the end point x = a Case III Test of continuity at x = b, if LHL = f(b) Then, f(x) is continuous at right end x = b. Max ([x],|x|) is discontinuous at

Let y = f(x) be defined in [a, b], then (i) Test of continuity at x = c, a lt c lt b (ii) Test of continuity at x = a (iii) Test of continuity at x = b Case I Test of continuity at x = c, a lt c lt b If y = f(x) be defined at x = c and its value f(c) be equal to limit of f(x) as x rarr c i.e. f(c) = lim_(x to c) f(x) or lim_(x to c^(-))f(x) = f(c) = lim_(x to c^(+)) f(x) or LHL = f(c) = RHL then, y = f(x) is continuous at x = c. Case II Test of continuity at x = a If RHL = f(a) Then, f(x) is said to be continuous at the end point x = a Case III Test of continuity at x = b, if LHL = f(b) Then, f(x) is continuous at right end x = b. Number of points of discontinuity of [2x^(3) - 5] in [1, 2) is (where [.] denotes the greatest integral function.)

If the function defined by : f(x) = {:{(2x-1,x 2):} is continuous at x =2, find the value of 'a'. Also discuss the continuity of f(x) at x =3.