How many of the following functions are continuous in all Real numbers? y = |x| y = sgn(x) 6 - y = Vx y = x? – a3 – a4 y = x + 5 O A) 5 O B) 4 C) 2 D) 3 O E) 1

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: Find the interval that the fullowing function is continous at In (9 -x) %3D A. [I,3] BL) (3,3) (1,3)…

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Q: What is the domain of the following function? f(z,9) 4-2 - y - In(y - z) + sin(z) +y 24 and y> =)…

A: Domain.

Q: Show that if u =x - y and v =r+ y, then for any continuous f, e-2s* f(x - y,x + y)dydr = || (a+n)e-…

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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: If f' is continuous on [a, b], show that 2 f(x)f'(x) dx = (F(b))? – (F(a)² %3D -

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Q: If f & R[a, b], then f is not continuous on [a,b].

A: Given statement statement:-  If  f∉Ra, b, then  f is not continuous on a, b.

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: Suppose that f(1) = 2, f(4) = 7, f'(1) = 5, f'(4) = 3, and f" is continuous. Find the value of *…

A: Use the method for integration by parts to solve the integral.

Q: Is the following bounded or unbounded and why? f(x,y) = ln(x2 + y2)

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Q: If f(x) is piecewise continuous on the interval (a,b), then J f(x) dx <oo. O (A) True. O (B) False.

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Q: 3. Use the definition to show that the function f(x) = is continuous on the interval [c, o0), where…

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Q: Where is f(x) = x-5 x² –6x+5 discontinuous? f (x)is continuous everywhere o 5 o 1 о 5, 1

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Q: For what values of a is

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Q: For what values of a and b is ax + 2b, g(x) = {x + 3a - b, 0 2 continuous at every x?

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Q: For what values of b is - b b + 1' g(x) x? + b, x > 0 continuous at every x?

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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: 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: Assume that the function f is such that f(3) = 1, ſ f(x) dx = 5, and f'(x) is continuous on [0, 3].…

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Q: 3.Show, by means of the example f(t) = sin t, that f ∗ f is not necessarily nonnegative.

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Q: As the fun ction Leay) xy js Continuous I+ x*+y? e (-1, 2)

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Q: Let f(z, y) be a continuous function such that f2, Su Jay and fuz are all continuous everywhere. If…

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Q: For what value of b is

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

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Q: Let f and g be continuous funCHons such that So f(x) dx =9,S f(x) dx = 5, and S g(x)dx= -7 what is…

A: Given, ∫06fxdx=9∫36fxdx=5∫30gxdx=-7

Q: For what values of a is Ja'x – 2a, x 2 2 f(x) : \ 12, x < 2 continuous at every x?

A: Given function is Since f(x) is continuous for every x. So, f(x) is continuous at x=2.

Q: 45. For what value of a is x2 - 1, x < 3 2ax, f(x) = %3D 2ах, continuous at every x?

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Q: If f(3) = 15, f' is continuous, and I f'(t) dt = 24, what is the value of f(4)? %3D

A: In this question, f is a primitive function and we have to find f(4). So I am providing the…

Q: Let f be a continuous function such that | f(x) dæ = 5 and f(x) dx = 3. %3D | 4f(æ) dæ? What is the…

A: Recall: Properties of integral:  i)   ∫abf(x)dx=-∫baf(x)dxii) ∫abf(x)dx+∫bcf(x)dx=∫acf(x)dxiii)…

Q: Suppose f: a,b] → R is continuous and faf=/ for all xie [a,bl Show that f(x)= 0 ft rEla,b).

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Q: What values must a and b take for f to be continuous everywhere?

A: Given that the function fx=ax1+4x2 ,      x&lt;0b    ,                     x=0tan-1xx ,…

Q: Use the definition to show that the function f(x) = → is continuous on the interval [c, o), %3D…

A: The given function is f(x)=1x2 .we have to show that f(x) is continuous on the intervel [c,∞)…

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: Suppose that f e H(C) and |f(2)I < eRez for all z. Show that f(z) ce for some constant c.

A: Consider fz=cez for z=x+i y Take absolute value on both sides: fz=cez=cex+iy=cexeiy=cex…

Q: Which of the following interval is f(x) = O (2, +∞0) O [2, +∞0) 0 (-∞0, +∞0) ○ (0, +∞) 1 Vx…

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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: 5 Show that the function h(r, y) defined by if r +y? < 1 h(x, y) 0; if r +y 2 1 is discontinuous.

A: For continuity of 2 variable function L.H.L=R.H.L=Function value at (x,y). limx→a+,y→b+…

Q: Find the values of a and b that make f continuous everywhere: x²-4 if x 3 -

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Find c such that the function 10 I C - is continuous everywhere.

A: A function f(x) is said to be continuous at a point x=a if limx→af(x)=f(a). That is, a function f(x)…

Q: If f' is continuous on [a, b], show that 2 [* f(x)f'(x) dx = [f(b)J* – [f(@}J°

A: Proof:            Given: f' is a continuous…

Q: Find the value of a so that the function is continuous everywhere x2 + ax – 2, x 1 O 1 O 2 O 3 O 4 O…

A: Given:   f(x) = x2+ ax - 2  when x≤ 1     f(x) = x + 3 when x &gt; 1

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

A: Given that f is continuous on a,b. ∫abπ fx dx = π ∫abfx dx Say that the statement is true or false.…

Q: Suppose f is differentiable on (a, b). If c E (a, b) and f' (c) = 0, thenf (c) is either the maximum…

A: Given f is differentiable on (a,b). and c lies in (a,b) such that f'(c)=0.

Q: . For what value(s) of c is f(x) = 54-x³, x < 3) %3D lcx² + 13, x 2 3) continuous everywhere?

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Q: For what values of a and b is ax + 2b, g(x) = {x? + 3a - b, 0<xS2 3x 5, %3D - continuous at every x?

A: The given function is gx=ax+2b,        x≤0x2+3a-b,  0&lt;x≤23x-5,           x&gt;2. For the given…

Q: Assume that is a continuous real function on (a, b) such that 1 1 o ( x + ³) ≤ ²}/20 (x) + = 0 ( for…

A: Here given that φ is a continuous real function on a, b such that φx+y2≤12φx+12φx

Q: If f(x) is continuous with f(a)f(b) < 0, then the e Select one: O an indeterminable number of roots…

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Q: If f:R → R is continuous at both 2 and 4, then f must be continuous at some c E (2,4). Select one:…

A: Consider the function f:R→R, that is continuous at both the points 2 and 4. Let there be a point c…

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.