3. Let f be a bounded continuous function on C and a E C. Show that f(z) dz lim 0. iz]=r (z – a)² r→∞ |

Q: 2. Show that f(r) = sin r is uniformly continuous on (0, 00). %3D

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A: The given question has been solved in detail

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Q: Show that the function f defined by (x, y) = (1, –1) f(x, y) = r? + y (x, y) # (1, –1) x + y is not…

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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: If f is continuous at (0, 0) and f(0,0) = 10, then lim f(x, y) = 10. (z,y)-(0,0)

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Q: Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = x2 + Y (x, y) # (1, –1) x +y is…

A: Here, the given function is: f(x,y)=1;   (x,y)=(1,-1)x2+yx+y; (x,y)≠(1,-1) lim(x,y)→(1,-1)

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Q: f(z) =|z| is continuous in C 0 صواب ihi O

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Q: Let if (x, y) # (0, 0) f(x, y) = { if (x, y) = (0, 0). (a) Compute the limit as (x, y) → (0, 0) of f…

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Q: Show that the function f defined by (x, y) = (1, –1) f(r, y) = { x2 + y (x, y) # (1,–1) r + y is not…

A: To show that the function f defined by fx, y=1, x, y=1, -1x2+yx+y, x, y≠1, -1is not continuous at 1,…

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: Let f : [0 , 1] → R be a continuous function on [0 , 1] such that : f(x)x" dx = 0, (n = 0, 1, 2,…

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Q: (3) Determine the values of a for which x4 + y* , if(x, y) # (0, 0), (x² + y?)ª f(x, y) = ,if(x, y)…

A: solution.

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Q: Show that f(z) =xy+iy is everywhere continuous but is not analytic

A: Answer

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A: Given function is  fx,y=1,             x,y=1,-1x2+yx+y,    x,y≠1,-1

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Q: Show that f is continuous at (0, 7) where a cos y + y(e" – 1) if r + 0 f(x, y) = ay if æ = 0.

A: In the given question we have to show that the given function f(x,y) is continuous at the point…

Q: Show that the function f defined by 1, (x, y) = (1,–1) f(r, y) = { x² + y (x, y) # (1,–1) r+ y is…

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Q: I. Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = x2 + y (x, y) # (1, –1) x +y is…

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Q: Define f(0,0) in a way that extends f to be continuous at the origin. 6x² -x²y² + 6y² x² +y? f(x,y)…

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Q: Let F be defined on R x R to R by F(x, y) = x² – y. Show that F is C' on a neighborhood of (0, 0)…

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Q: Show that the function f defined by 1, (x, y) = (1, – 1) f (x, y) : .2 x + y (x, y) # (1, – 1) x + y…

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

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Q: Determine the absolute extremes of the function f(x,y)=x²+2y²-x over the closed and bounded set R…

A: By using concept of absolute maxima and minima in it's domain.

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Q: (a) Let Ty if +y +0, g(x, y) = 0 if r = y = 0. Prove that g(r, y) is continuous at (0,0).

A: To Prove: g(x,y) is continuous at (0,0)

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A: In this problem, we have to check condition for having the function continuous.

Q: I. Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = { x2 + y (x, y) # (1, –1) x + y…

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Q: Consider the function { that is defined on R as: f(x)=-x² +4x-3. Designate by (C) its representative…

A: The given function is, fx=-x2+4x-3 The orthonormal system is, O; i→,j→ To evaluate limx→-∞fx  and…

Q: 2. Let f: I→R be a continuous function on a bounded, closed interval I= [a, b). Show that f has a…

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Q: (a) Assume that g(x, y) is a function of the form 2+y² g(x, y) = 0 if (x, y) = (0,0). (i) Show that…

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Q: Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = { x2 + y (x, y) # (1, –1) x + y is…

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Q: 1. Suppose f(r) is a continuous function on r € [a, b]. Prove f(x)dr < S°S(x)| dx.

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Q: 3. Evaluate the limit r(t +h) - r(t) lim h→0 h for r(t) = (t-1, sin(t), 4).

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Q: LET 2.3 + xy? if (x, y) # (0,0) f(x, y) = x² + 2y4 if (x, y) = (0,0). Show that f is not continuous…

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Q: 3. Let f(r) = r sin() if z #0, and f(0) = 0. Show that f is continuous at r= 0 by using e- d…

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