Let f : X 'n Y be a continuous function. If X is disconnected then so is f(X). True False

Q: If f is differentiable at a , then f is continuous at a . True O False

A:

Q: If f is continuous on A) True B False on [a,b], then Sº [ f(x a xf(x) dx = x f(x) dx

A: It will solve by part method integration  ∫u v dx = u∫v dx −∫u' (∫v dx) dx u is the function u(x) v…

Q: Let f be a continuous function such that f(x) dx: = -1 and f(x) dx = = 9. %3D 2 2 | f(x) dæ? What is…

A:

Q: Let Assume that ƒ has a continuous derivative on [a, b]. Show that there exists a polynomial p(x)…

A: Given f has continuous derivative on a,b then for all ε>0 there exist a polynomial qx such that…

Q: Is f(x)= Vx + 1 continuous at x == -1

A: Draw the graph and check continuty with help of limit of function.

Q: If f is continuous at a number, x, then f is differentiable at x. True

A:

Q: Find an example of a function f:[0,1]¬R such that f is not continuous, but f does satisfy the…

A:

Q: f(x+y)=f(x)+f(y) and f is continuous at 0. Prove that f is continuous at any a.

A: To show that the continuity at x=0 implies continuity at all x (under the given condition on the…

Q: Is the function continuous on the interval (0, T COS x O no O yes O cannot be determined

A: Given problem:- Is the function ex/cosx continuous on the interval [0, π] ?

Q: Suppose that f(x) is a continuous function on [a , b] and F(x)= | f(t) dt a Part 1 of the…

A:

Q: If f(x) is continuous on the interval [a, b] and a <c < b, | f(x) dæ is equal to then

A:

Q: 10 If f is a continuous function and F'(x)=f(x) for all x € R, then x)dx=?

A:

Q: Theorem 11. Let f be a continuous function on R such that f(x+y)-f(x)+fv) vx, yER. Show that f(x)3cx…

A:

Q: Answer true or false. If f is not piecewise continuous on  [0, ∞>),  then  ℒ{f(t)}  will not exist.

A: Given problem:- If f is not piecewise continuous on [0, ∞),then ℒ{f(t)} will not exist.  True…

Q: If f is continuous on [a, b] and "If(x)| dx = 0, what can you conclude about f ?

A: Solution:Given If f is continuous on a,b and ∫abf(x) dx = 0To conclude about f ?

Q: Let f(x) be a continuous function on R-{0} such that (f(x) dx exists for all a > 0.If g(x)= |- dt, t…

A:

Q: If f is differentiable at a number, x, then f is continuous at x.

A:

Q: If f(x) is continuous at x = c then f'(c) exists. True False

A:

Q: Suppose f" is continuous everywhere. For any constants a and b, show that xf" (x) dx = bf" (b) –…

A: Explained below

Q: Prove that the function f(x) = 2x is uniformly continuous on R.

A:

Q: Prove that if f :V → W is differentiable at a point a E V it must be continuous there.

A:

Q: The function f is a composite of two continuous functions: y = cos x and y = x 2, so f is continuous…

A: The function fx is a composition of functions y=cosx and y=x2and it is a continuous function.

Q: Suppose that f (x) &lt; 0 if x is positive and f (x) &gt; 1 if x is negative. Can f be continuous at…

A: f(x) will be continuous at x = 0 if LHL = f(0) = RHL

Q: Use Theorem 9.2.1 to show that T is not continuous at 0, no matter what value is chosen for b.

A:

Q: Suppose that f(a) = f(b) = g(a) = g(b) = 0 and the second derivatives of f and g are continuous on…

A:

Q: Show by definition that f(x) = |x| : R → R is continuous on R. 2.

A:

Q: If f is continuous everywhere except at x = 2, then f(x +1) dx is not integrable. True False

A: To check whether the given statement is True or False."If f is continuous everywhere except at x=2,…

Q: Prove that f(x) =√x is uniformly continuous on [0,∞).

A: According to the question, we have to prove that the function fx=x is uniformly continuous on [0,∞).…

Q: If f : R+ → R is a function such that f(x + y) = f(x)f(y) for all x,y ∈ R+. Prove that f is…

A:

Q: Let f : X → Y be a continuous function. If X is disconnected then so is f(X). True False

A: Let f : X → Y be a continuous function. If X is disconnected then so is f(X).

Q: Prove thait if y=f(x) f(x)=0 when x is a rotional number, then f(x)=0 for all x € R. is continuous…

A:

Q: If a function f(x) is not defined at x-c, then it is not continuous at a close interval [a, c). * O…

A: We know that function is continuous at c if the 1) f(c) is defined 2)limit of f(x) exist, Here…

Q: is continuous and g and h are differentiable functions, show that ch(x) d (e f(t)dt f(h(x))h'(x) –…

A: In the given question, the concept of the Fundamental Theorem of Calculus is applied. Fundamental…

Q: . Explain why f (x) = sin (x² + 3r + 1) is a continuous function for all c eR.

A:

Q: Let f(x) = CX– 2, x > I lx+ 8, x < 1 For what value of c is f(x) continuous on (-00, 0)

A:

Q: Let f be a continuous function on R such that f(x+y)=f(x)+(y) ¥x, yER. Show that f(x)3cx for some…

A:

Q: Use limits to determine if f(x) = 1 is continuous at x = 0. (z+2)?

A:

Q: Suppose that f(1)=8, f(4)=−4, f′(1)=−3, f′(4)=2, and f′′ is continuous. Find the value of ∫from 1 to…

A:

Q: Show that the following function is not continuous at (0,0). { 3 x2+y6, 0, , if (x, y) # (0,0); if…

A: To prove that given function is continuous. gx,y=xy3x2+y6,   if x,y≠0,00,              if x,y=0,0

Q: Let f: RR be a function satisfying f(x + y) = f(x) + f(y) for all x, ye R. If the function f(x) is…

A: Note :   Let r&gt;0 and c∈R ,then vr(c)={x∈R : x-c&lt;r }=(c-r ,c+r) ,which is called r…

Q: Let f be the function defined by: sin x, x 2 - For what values of x is f NOT continuous?

A: Given function is: fx=sinxx&lt;0x20≤x&lt;12−x1≤x&lt;2x−3x≥2

Q: Let 'f be a continuously differentiable real valued function defined bx+a x<1 by f(x) = 5x² Then the…

A:

Q: show that f,given by f(x) x +0 )is continuous on R-{0}

A:

Q: If f is a continuous function on [a, b], ƒ is differentiable on (a, b), and if both f(a) and f(b)…

A: This question is based on Rolle's theorem which is explained in step 2.

Q: Theorem 12. Let f be a non-zero continuous function on R uch that f(x+y)=f(x) f(y) ¥x, yER. Show…

A:

Q: Find the value/s of a such that the function f is continuous everywhere.

A:

Q: Let g(x) be a continuous function for all æ E R. If f'(x) = g(x)f(x) for all a E R, find f(x). Note:…

A:

Q: . Is f(x)= Vx + 1 continuous at x == -1? Yes, because all functions are continuous No, because the…

A:

Q: Show that f : (0,∞) → R defined by f(x) := 1/x is not Lipschitz continuous.

A: Definition:  Let S⊂ℝ and a  function f:S→ℝ is Lipschitz Continuous at x∈S if there is a constant C…

Q: Find a value of k making h(x) continuous on (0, 5). 0 < x < 2 2 < x < 5 h(x) = x +1,

A: A function is continuous if limx→a+ h(x)  = limx→a- h(x) = h(a) Now in the given question, function…