If f(z) is continuous at z, then it must be differenttiable at z, ylgn O İhi O

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

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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: If f is continuous function on la, bl, then zf(z)dz = = (z)dz. Select one: a. True b. False

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

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

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

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

A: Answer: False.

Q: Show that a function f : R? → R defined by S (7, y) = ! if (x, y) + (0,0) 0, if x = y = 0 is…

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Q: For what value of b is X - b b + l' x 0 continuous at every x?

A: The given function is gx=x-bb+1if x&lt;0x2+b if x&gt;0. If the above function is continuous at x=0,…

Q: Prove that f(z)=|z| is nowhere differentiable in C but is continuous everywhere.

A: Given function is fz=z. We have to prove that the function fz is nowhere differentiable in C but is…

Q: Let h: R → R be a continuous function on all of R. If h(x) T, prove that h(T)<0.

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

A: Explained below

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

A: Given: Lets consider the given, that Lft will not exist if f is not piecewise continuous on [0,…

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

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Q: f(z) = Im z is continuous

A: Let us use the convergence criterion to prove that Im z is continuous. Let us assume that  znn=1∞ be…

Q: Suppose f is differentiable on some open interval containing e and f" (c) = 0 then (c, f(c)) is a…

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

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

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Q: Prove that if f (x) is continuous at x = a and g(y) is continuous at y = b, then F(x,y) = f(x)g(y)…

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

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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: iii. is continuous at a provided g(a) # 0.

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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…

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Q: Iff is continuous at a point a and c e R, then cf is continuous at a.

A: Note that, f is said to be continuous at 'a' if limx→af(x)=f(a) And, (cf)(x)=cf(x)

Q: Let h: R → R be a continuous function on all of R. If h(x) T , prove that h(T)<0.

A: Given that, let h:ℝ→ℝ be a continuous function on all of ℝ. Also given, hx≤0 for all x&gt;π.

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

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

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Q: 0. If y = Ax) is continuous at x = xo, and z = yo =Axo), prove that z = g[f{x)] is continuous at…

A: Explanation of the answer is as follows

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

A: I have proved limiting value is equal to functional value

Q: If f is continuous on [0, 1] such that x* f(x) dx=0, for n0, 1, 2, 3,.. then show that f(x)=0 on [0,…

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Q: Show that the limit of the function f(x,y) = at (0,0) does not exist. (x+y)² x² + y²

A: Limit of a Function: Like a single-variable function, a multi-variable function can also have limit.…

Q: Suppose f and g are differentiable functions on R such that,

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Q: If x is ar.v. with c.d.f. F(x), F(x) is continuous at x = 3, then p(x = 3) = 0 %3D O True False

A: It is given that the CDF of x at x = 3 is continuous.

Q: Let f, g : D → R be continuous at c E D. Prove that fg is continuous at c.

A: Let ε&lt;1 and M= maxf(a), g(a), 1, then there exists δ&gt;0 such that f(x)-f(a)&lt;ε3M (since f is…

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

A: For the solution of the problem follow the next step.

Q: Suppose that f : R →R is continuous and that its image f(R) is bounded. Prove that there is a…

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Q: Let f(x) = 0 if x∈Q if x and f(x) = 1 if x ∈R\Q. Prove that y=f(x) is not continuous at any point…

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Q: suppose f (x) = Vx (x>0),prove that f is differentiable (0,00) and find f'(x) on

A: Function f(x) = √x (x&gt;0)  We know f(x) =Xn

Q: prove that if f(x)is derivable at a point say "a" then it is continous at x=a

A: Here we have given that the function fx is diffrentiable at a point  .x=a that is f'x exists.…

Q: Prove that if m(x) is differentiable on (−∞,∞) and its derivative m'(x) is bounded then m is…

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Q: Suppose that f is a continuous function on R and, g and h are differentiable functions on R.…

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Q: Suppose f is a continuous on R and f(x + y) = f(x) + f(y) for all r,y R. Prove that for some…

A: What is Continuous Function: In mathematics, a continuous function is one that, in response to a…

Q: If f(z) is continuous at z then it must be differenttiable at z, 0.

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

A: Solution

Q: Suppose f ≤ 0 , f is continuous on [a, b] and ∫ab f dx= 0. Prove f(x) = 0, ∀ x ∈ [a, b].

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Q: If x is a r.v. with c.d.f. F(x), F(x) is continuous at x = 10, then P(X= 10) = 0. True False

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Q: Suppose f is continuous and F'(x) = f(x). J s(x)dx =3, f(x)dx = 4, [f(x)dx= 5, and F(1) =6, evaluate…

A: Here this is simple application of integration and differentiation please watch carefully.

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: