Suppose f is a function defined and continuous on the open interval (a, b). Suppose g is a function defined and continuous on the open interval (a, b). a) Make up an example with f, g, and (a, b) that shows that does not have to be continuous on (a, b).

Q: The set of functions {sin nx}, n = 1, 2, 3, ... , is %3D orthogonal on the interval [ -n, n]. Show…

A: Given- The set of functions sin nx, n = 1, 2, 3, ⋯, is orthogonal on the interval -π, π To Show-…

Q: A continuous function defined on an open interval (a,b) must have a minimum value. True False

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Q: Let f be continuous from [0, 1] to [0, 1]. By posing 6(x) = f(x) – x, show that fhas a fixed point…

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Q: Verify Mean Value Theorem for the function f(x) = J in interval (0,9]

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Q: Verify Mean Value Theorem for the function f(x) = in interval [0,9]

A: If a function  f(x) is continuous at the range [a, b] and also differentiable in the range (a, b)…

Q: What are the Bounded functions?

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Q: Sketch the graph of a function f  that is continuouson [1, 5] and has the given properties.   f has…

A: f is continuous on [1,5]. f has no local maximum or minimum. 2 and 4 are the critical numbers.

Q: Sketch the graph of a function f that is continuous on [1, 5] and has the given properties.…

A: Given data: The absolute maximum at MA=5. The absolute minimum at mA=2. The local maximum at MB=3.…

Q: Prove there does not exist a continuous function which maps the closed interval [0, 1] onto the open…

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Q: Show that f (x) = is not uni form continuous on (0,3) -3

A: To prove that 1x-3 is not uniformly continuous on 0,3

Q: Define a weakly quasi-concave on a Cobb-Douglas Function

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Q: Show that the functions are continuous on their natural domains with examples?

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Q: With X a topological space, is the function defined as A:X →X × X,x → (x, x) continuous? Investigate

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Q: Sketch the graph of a function f  that is continuouson [1, 5] and has the given properties.…

A: Sketch the graph of a function that is continuous defined on the interval [1,5] having the following…

Q: Let f be the function defined byf (z) = 3z2 -z. What is the absolute minimum value of f on the…

A: Topic = Application of Derivative  The absolute minimum value of f is 0.

Q: Sketch a graph of a function f that is continuous on (-0,0) and has the following properties. f'(x)…

A: Given that: f'x<0 and f''x<0 on -∞,3; f'x<0 and f''x>0 on 3,∞. We know that if, f'x>0…

Q: What is the function f on (a, b) if for all x1, X2 in (a, b) when x, f(x2). is it decreasing or…

A: for all x1 , x2 in a,b when x1<x2 then fx1>fx2 claim- is it decreasing or increasing

Q: 7. has a fixed point. ) Prove that if f is a continuous function from [2, 5] to [2, 5], then f

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Q: Show that f is continuous on (∞ , -∞ )  .

A: we have to show that f(X) is continuous in (-∞,∞). f(x)=x2             if   x<1           x…

Q: fully continuous for all x E R by use of the product rule differentiate the function f{x} = x3e-4x

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Q: If f(z) is analytic function show that:

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Q: Find four relations from {a,b} to {x,y} that are not functions from (a,b} to x, y}

A: We have to find four relations from {a, b} to {x, y} that are not functions from {a, b} to {x, y}.…

Q: Show that the following function is not continuous at (0,0). { 3 x2+y6, 0, , if (x, y) # (0,0); if…

A: To prove that given function is continuous. gx,y=xy3x2+y6,   if x,y≠0,00,              if x,y=0,0

Q: Assume that f is a function with domain [0,1] that is differentiable in (0,1). Assuming that f (0) =…

A:

Q: Show that the function is continuous on its domain. State the domain. Vx + x cosx

A: fx=x4+x3 cosx The domain is the set of all positive real numbers including zero. That is, the domain…

Q: Find four relations from {a, b} to {x, y} that are not functions from {a, b} to {x, y}.

A: It is given that the two sets are {a, b} and {x, y}. It is known that if A and B are the sets then…

Q: Let f:C-R be a function where Cis a closed and bounded interval. Prove that if f is continuous then…

A: The given problem is to prove that the function is maps from closed and bounded interval to Real…

Q: if f functions is bounded on [a,b] does f has an absolute maximum and absolute minimum or not??

A: if f functions is bounded on [a,b] does f has an absolute maximum and absolute minimum or not??

Q: Every continuous function on a, b, that has absolute extrema on endpoints, must have one critical…

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Q: Suppose that the functions f and g are defined throughout an open interval containing the point x,,…

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Q: Suppose that f: R² R is a function given by y³ x² + y² f(x, y) = 0 for x = y = 0. Show that f(x, y)…

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Q: Show that an isometric function is Uni Formly continuous and ce-to - one

A: Definition: Let X and Y be metric spaces with matrices dx and dy , A map f : X→Y is called an…

Q: Show that f (x) =++1 1 - is not uni form continuous on (-1,0)

A:

Q: Let ƒ be a function that is continuous on [0, 2] and đifferentiable on (0, 2). Suppose that f (0) <…

A: Yes, we will show that f' necessarily take on both positive and negative values on (0, 2). If the…

Q: Consider a function f that is continuous on [-2, 4] with f'(x) = 0 whenever x = 3 or x = 5. Suppose…

A:

Q: Determine whether the statement " If f is continuous for all x and y and has two relative minima,…

A: Given: If f is continuous for all x and y and has two relative minima, then f must have at least one…

Q: 5. Show that the functions x,x? -,x³ -x x are mutually orthogonal in the interval [-1,1].

A:

Q: Suppose f is a bounded real function on [a, b), and f' eR on [a, b). Does it follow that fe R? Does…

A:

Q: Determine the set of points at which the function is continuous

A: f (x, y, z) = sin-1 (x2 + y2 + z2) = t (Let's say)Hence, sin t = (x2 + y2 + z2)

Q: Suppose f(x) is a continuous on the interval [a,b] and c is a number in (a,b). If f"(c) is…

A: If f(c) is a local maximum then f''(c) &lt; 0 and if f(c) is a local minimum then f''(c) &gt; 0

Q: prove: f(x)=5x is continuous at x=2     note: please prove this in theorem format(two columns)

A: Consider the given function. y=5x Now, check the left first.…

Q: estimate the type of the function at the interval {(1,2), (2,5), (3,5), (4,7)} is it: 1-1 or onto..?

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Q: 8) Assume that 2 + Va if x >1 if x < 1 f(x) = Determine whether or not f is continuous at x = state…

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Q: Sketch the graph of a function continuous on the given interval that satisfies the following…

A: Given, f is continuous on -∞,∞; f'(x)&lt;0 and f''(x)&lt;0 on -∞,0; f'(x)&gt;0 and f''(x)&gt;0 on…

Q: Give an example of a function that is integrable on the interval [−1, 1] but not continuous on [−1,…

A: Let fx=x       Greatest integer function. Now, ∫-11x dx=∫-10x dx+∫01x dx=∫-10-1 dx+∫010…

Q: Find four relations from {a,b} to {x,y} that are not functions from {a,b} to {x,y}.

A: Given: Let A=a, b and B=x, y To Find: Four relations from A to B that are not functions.

Q: Sketch the graph of a function f that is continuouson [1, 5] and has the given properties. Absolute…

A: Given: The function is continuous on [1,5] The absolute minimum is at 2 The absolute maximum is at 3…

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.