If f is continuous at a then [f ,a]=lim x->a f(x)-f(a)

The contrapositive of statement: If f(x) is continuous at x=a then f(x) is differentiable at x=a

If a continuous function f on [0, a] satisfies f(x)f(a-x)=1, a >0, then find the value of int_0^a(dx)/(1+f(x))

If a continuous function f on [0,a] satisfies f(x)f(a-x)=1,agt0 , then find the value of int_(0)^(a)(dx)/(1+f(x)) .

If f(x) is continuous in [0,2] and f(0)=f(2). Then the equation f(x)=f(x+1) has

Let f(x) ={{:( xe^(x), xle0),( x+x^(2)-x^(3), xgt0):} then the correct statement is (a) f is continuous and differentiable for all x (b) f is continuous but not differentiable at x=0 (c) f is continuous and differentiable for all x . (d) f ' is continuous but not differentiable 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

If f(x) is continuous in [0, 1] and f((1)/(3))=12 , then the value of lim_(nrarroo)f((sqrtn)/(3sqrtn+1)) is equal to

What happens to a function f(x) at x=a , if (lim)_(x->a)f(x)=f(a)

If f:RrarrR is a continuous function satisfying f(0)=1 and f(2x)-f(x)=xAAxepsilonR and lim_(nrarroo)(f(x)-f(x/(2^(n))))=P(x) . Then P(x) is

If f(x) is a continuous function such that f(x) gt 0 for all x gt 0 and (f(x))^(2020)=1+int_(0)^(x) f(t) dt , then the value of {f(2020)}^(2019) is equal to