Let f:XT, → XT, such that f({a}) = {b, c} į Is fa continuous function? i yes,no, or can't you tell? Let g:XT, → Xr,g({b}) = {b, c} i Is ga continuous functionm? i yes, no, or can't you tell?

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: Theorem 7. Let X be a continuous RV, g a function from R to R. Let Y = g(X). Then E[Y] = E[g(X)] = /…

A: The expected value of random variable is the weighted average of all possible values of the…

Q: Isin , it (z, y) # (0, 0) 2. Determine if a function f : R → R defined by f (r, y) = 0, if (z, y) =…

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Q: Let f : [0, 1] → R be a continuous function. Show that the function g : [0, 1] → R given by g(x) =…

A: The given function: is continuous. That is, assume f(x)=x, then x is Real number. Show that the…

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…

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Q: x² - y² z = g(x, y) = x² + y² 0 for for (x, y) = (0,0) (x, y) = (0,0)

A: Given function is z=gx,y=x5−y5x4+y4 for x,y≠0,00            for x,y=0,0. We have to check whether…

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: 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: A function F is defined on R² By: f (x, y) (x, y)# (0,0) F (x, y) = %3D (x, y) = (0,0) x²y+2xy² Let…

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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: Consider the following function of two variables: xy g(x, y) = x² + y? Prove that the above function…

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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)

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: 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: (a) Assume that g(x, y) is a function of the form g(x, y) = {" 0 if (x, y) = (0,0). (i) Show that…

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Q: 2³ – a²y – y³ x² + y² - si (x, y) # (0,0) g(x, y) = si (x, y) = (0,0)

A: NOTE: According to guideline only one proof can be given, so proof of continuous is given here, for…

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

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Q: 2.) Determine the set of points at which the function is continuous. а. F(x,у) — 1+x²+y² 1-x2-у? ху…

A: Here, to find the set of points at which the function is continous.

Q: H.W-2:Is the following functions continuous or not at z = 0? %3D Re(z) f(z)= 1+ z %D Z

A: Topic:- calculus

Q: What value of c makes the following function continuous at (0,0)? f(z, y) = C, x+ y³ +2, (x,y) +…

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Q: Let f : (a, b) → R be continuously differentiable, where −∞ ≤ a 0 and a C > 0 such that |f(x) −…

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Q: Let h(x) = f(g(x)), where f and g are differentiable on theirdomains. If g(1) = 3 and g′(1) = 5,…

A: Given: Let h(x) = f(g(x)), where f and g are differentiable on theirdomains. If g(1) = 3 and g′(1) =…

Q: Determine if the function √x² + y² +1 x² + y² f(x, y): 1 2 is continuous at (x, y) = (0,0). If not,…

A: To find whether the given function is continuous or not at(0,0).

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,…

A: Given that f:0,1→R be a continuous function on 0,1 such that ∫01f(x)xndx=0. The objective is to show…

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: Let f be a continuous (and therefore uniformly continuous) function on [a,b). Show that, given e> 0,…

A: Consider the given information.

Q: x³ - y³ z = f(x, y) = x² + xy + y² 0 for for (x, y) = (0,0) (x, y) = (0,0)

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

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Q: Prove that there is a unique continuous function y : [0, 1] → R such that y(r²) x E (0,1]. y(x) = e"…

A: Differentiate the equation wrt x:  ddxyx=ddxex+yx22y'x=ddxex+ddxyx22y'x=ex+xy'x2

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 ( x, y ) = ln ( ex * Y ) be a function. At which point the function is discontinuous? O a. (…

A: Given: f( x, y) = ln ( ex+y )     To determine: The point at which function is discontinuous.

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: fhow thot te function f depined by f(XrY) = 1 (x,Y) =(1,-リ - 1) %3D x2 +y (x, y) F (I,-1) * t y is…

A: We use the sequential definition of continuity to solve this problem.

Q: Show that the function F: C[0, 1] → R defined by F(f) = max f(x) zE[0,1] is continuous.

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Q: Determine whether or not the following functions are continuous at (0,0). -1 for (x, y) = (0,0) a)…

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Q: 1. Suppose f(r) is a continuous function on r E [a, b]. Prove f(x)dx < Slf(x)| dx.

A: According to our guidelines we can solve only first question and rest can be reposted.Thank you

Q: If f is continuous on [a,b], then the function g defined by g (x) = S f (t) dt, a < x < b is…

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Q: Show that a function f : R2-R defined by f(x,y)= x3- y3 / x2+ y2, if (x,y) is not equal to (0,0)…

A: What is Continuous Function: A function is referred to be continuous at a certain point in its…

Q: At what points (x,y,z) in space are the functions continuous? a. h(x,y,z) = In (5z° - x-y² - 3) 1 b.…

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

Q: Let f be a continuous function with f(x) > 0 for all æ € R. Define r(z²+1)" g(r) = e" f(t)dt.

A: Let f be a continuous function with fx&gt;0 ∀x∈ℝ. A function g is defined by  gx=∫0x2+1xexftdt…

Q: Show that the function f defined by 1, f(r, y) = { r2 + y (x, y) = (1, –1) (x, y) # (1, –1) is not…

A: We have to show that  f(x, y)  is not continuous at  (1, -1)

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: Given function is fx,y=1,           x,y=1,−1x2+yx+y  ,x,y≠1,−1 . We have to show that the given…

Q: Let f(x) be continuous on [0, oo) and twice differentiable on (0, co). If f'(x) > 0 and f"(x) 1,…

A: Let f(x) be a continuous function on [0,∞) and differentiable at 0,∞. The given double derivative…

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: Show that the function f defined by (x, y) = (1, – 1) f(x, y) = 1² + y (x, y) # (1, –1) I + Y is not…

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Q: Let f : R → R be a continuous function that is convex on (-0o, 0 and 0, 00) and has a local maximum…

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