Q4) let f be non-negative continuous function of [a, o]& k> 0 is constant. IF t f(t) < k| f(s) ds, t> a holds, then f(t) = 0 for t > a

Q: Iベ+g2 Let f(x y) s (x,y) = (0, 0) . ノ Is f Continuous at 10,0) ? why ?

A:

Q: 2. Choose and work either a) or b), NOT BOTH. (a) Prove that the function f (x) = 1 is NOT uniformly…

A: To answer the given questions we shall use the sequential criteria for continuity and uniform…

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: B) Let f:[-2,2] R, then show that the function sin for x 0, f(x) = 0) for x 0, is uniformly…

A:

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…

A:

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

A: Let (X, ΤX) and (Y, ΤY)be topological spaces A function f : (X, ΤX) → (Y, ΤY) is continuous  if…

Q: 1. Assume f(r) is a twice continuously differentiable function on the closed interval (a, b), which…

A:

Q: 4. Choose and work either a) or b), NOT BOTH. (a) Show that the function f (x) = sin z for x €R is…

A: In the question given that we have to solve any one of them. Let us solve the first one.

Q: Suppose that f is a continuous function for which f(1)÷0 and f(t) dt=0. What can we conclude about…

A:

Q: 2. Consider the function of two variables f(r, y) defined by parts as ² + y? f(r, y) = { 2/r² + y² +…

A: We have given f(x,y)=x2+y22x2+y2+9-6   if (x,y)≠03                                   if…

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: Find all values of c such that f is continuous on (-0, 0). 1- x2, f(z) = { – x², x. C = (smaller…

A: Given function is, fx=1-x2   , x≤cx          ,x&gt;c  Since 1-x2 and x are continuous on the…

Q: Q1 (a) Sketch the graph of two continuous function g(x) = cos x and h(x) = on the same plane,…

A: Solution is here

Q: 3. Let z = yg(x2 - y²) be a differentiable function. Show that z(x,y) satisfies 1 дz 10z 7 - + - = х…

A:

Q: Q1. Let f: R² → R be defined by x3 + y3 r² + y² if (x, y) # (0,0) if (x, y) = (0,0). f (x, y) = (a)…

A:

Q: B) Let f:[-2,2]R. then show that the function xsin -) for x 0, f(x) = for x = 0, is uniformly…

A:

Q: 4. Suppose that F(t) is a differentiable function of one variable. Define the function z(x, y) = e'F…

A: z(x,y) ey F (exy )diff wrt xdzdx = ey F' ex/ye.1/yzx = ey.ex/yyF' (ex/y)and diff wrt ydzdy= ey…

Q: (b). Show that iff= u+iv is analytic in a region S and u is a constant function (i.e., independent…

A:

Q: 2. Determine the points where f is continuous where x + Y if (x, y) # (0,0) f(x, y) = x2 + y2 if (x,…

A:

Q: ery differentiable function y=f(x) defined

A: Given statement is: Every differentiable function y=f(x) defined on an open interval (a,b) has an…

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.

Q: 10. If f(x) and F(x) then f(2x)dx = are continuous functions such that F'(x) = f(x) on R, %3D %3D

A: We have to solve given problem:

Q: f: R → R be a continuous function and 3' (f(2x+ry))-f(x-ry) s1 nex,y E R, r-1 prove that f(x) is…

A:

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…

A:

Q: how (in terms of  − δ) that a function f : R 3 → R defined by f(x, y, z) = (2x + 3y + 4z) is…

A: Solution of given problem is followed by

Q: 1. Show (in terms of  ε - δ) that a function f : R3 → R defined by f(x; y; z) = (2x + 3y + 4z) is…

A:

Q: 4. If continuous function f:[0,1] U (2,3) → R is one-to-one, is f also continuous? 1

A:

Q: x-1, -1 sx< 0 0<x <1 x = 1 1<x< 2 2 <x<3 2r, f(x) = 1, -2r + 4, 0, Is f(x) is continuous at x = 1…

A: Given: f(x)=x2−1,−1≤x&lt;02x,0&lt;x&lt;11,x=1−2x+4,1&lt;x&lt;20,2&lt;x&lt;3 A function f(x) is…

Q: Determine if a function f : R2 → R defined by f (x; y) = { x sin(1/x2+y2) , if (x, y) ≠ (0,0) and 0…

A:

Q: 1. Construct a continuous function f: I R on a bounded interval I with the following property: f is…

A:

Q: 12. Consider x²y x4 + y? f(x, y) : for (x, y) # (0, 0) and zero otherwise. Is f continuous at (x, y)…

A: For the solution follow the next step.

Q: Q\ Use appropriate functions to guarantee that (). x sin if x # 0 h(x) = 0, if x = 0 is continuous…

A:

Q: (3) Let f : R –→ R be a differentiable function everywhere. Prove that if f (x) = 0 has n solutions,…

A:

Q: 3. (10 points) Determine the value of C such that if r +0 f(r) := V16 + 3x – 4 if r = 0 will be a…

A:

Q: Q1. (a) Find the points (x ,y, z) in space, where the function f(x, y,z) = /x² +y² -1 Is continuous.

A: Given :- The points x,y,z in space , where the function f x,y,z = x2 +y2 -1 is continuous.

Q: 1. If g(x) is a continuous function on [2 ,9] such that , g(x)dx = 25 and ,9(x)dx = Find value of…

A: Using the properties of definite integral, it can be written as follows.

Q: (5) Prove that if f is continuously differentiable on [a, b], then |f(a) – f(b)| a – b max f'(x)|…

A:

Q: Find the absolute extrema of the function on the closed interval. f(x) = cos TX, [] minimum (x, y) =…

A:

Q: At what points (x,.y.z) in space are the functions continuous? a. h(x.y.z) = In (4z - 5x - 3y - 4)…

A:

Q: 1. Show (in terms of -5) that a function f: [2, 7] R be defined by f(r) V+1 is uniformly continuous.

A:

Q: 1. Find if the function is continuous in the given points (x, y) = (0,0) (x, y) = (0,0) x + y = 0 x…

A:

Q: 3. Show that there does not exist a non-constant continuous function f : R2 → R such that f(x, y) =…

A: Given:- f: R2→R

Q: II. Find all values of a and b that make -а - х, f(x) = {bx² + 1, Vx + b, x<-2 -2 sx<1 x21…

A:

Q: 1 if z 0. Is F(x) is continuous at I = 0? Justify your answer.

A: Given that for x&lt;=0, F(x) = (x - 1)/4 Then as x tends to zero from the negative side of zero, we…

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…

A:

Q: Σ 1 is continuous function in R. Prove f(x) = 4 n=1 n' +x*

A: The given function is                                     f(x)=∑n=1∞1n4+x4 To show that the above…

Q: If f is positive-valued continuous decreasing function and lim - S(x) = 0 then J F(x)dx diverges…

A:

Q: Use the properties of continuous functions to explain why f(x) = x cos(x)+1 is a continuous function…

A: y=x and y=cos(x) both are continuous functions  So the product of two continuous functions is a…

Q: 6x2 - x²y +6y2 x2 + y2 Define f(0, 0) in a way that extends f(x, y) = to be continuous at the…

A: The given function is- fx,y=lnx2y-x Partially differentiating it w.r.t. x- ∂f∂x=2xy-1x2y-x Again…