Let fn RR be a sequence of continuous functions which converg uniformly to a function f : R → R. Let (xn) be a sequence of re numbers which converges to x E R. Show that fn(n) → ƒ(x).

Q: 3. Consider the function f(x) = sin x. (i) Is there a sequence of polynomials converging uniformly…

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Q: Let f(x) be a continuous function. If f(c) is an absolute max or absolute min, then c is unique. O…

A: Given,  fx is a continuous function fc is absolute max or absolute min , then c is unique.  We have…

Q: a delta - epsilon proof that shows that the function f(x) = |x| is continuous on its do

A: Definition:

Q: Let 4 – x² if x > 1 f(r) if x < 1 Determine whether or not f (x) is continuous at r = 1.

A: If left hand limit and right hand limit same, then f(x) is continuous at x=1.

Q: The function f(x) = 2 + 1/x is continuous on the following interval EXCEPT. O a. [-2,-1] O b. [1,2]…

A: At any point a for a function f(x) the limit limx→afx=L exists if the limx→a−fx=limx→a+fx=fa.

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A: To determining intervals where a function is increasing or decreasing, first find domain values…

Q: Suppose that f is a continuous function such that | f(x) dz = -12. Find | () dr.

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Q: 3. Consider f(x)=r. For which range of values will the function converge or diverge?

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Q: Let f: A-XX Y be defined as f (a) = (f, (a). f2 (a)) each a A, Then f is continuous if and only if…

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Q: Consider a continuous function f(x)=2ex−1−2x. The graph of f is decreasing on which interval/s?

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Q: The interval in which the function f(x) = 2x – 12x is decreasing is O a. (-0,0) O b. (-0,-4) O c.…

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Q: Let f(x) ƒ-¹(x) = = 1 x + 13

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Q: Show using continuity directly that the function ƒ(x) = x1/2 is continuous on (0,+∞). Use any…

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Q: x2+2x+5. Show, x+2 Q1 Let f: R\ {-2} → R be given by f(x) directly from the definition, that f is…

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Q: Decide if f(x) is uniformly continuous on the interval. Explain

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Q: f(x) = {*- Įx² +px + 2,x # 3 3, x = 3

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Q: x+2, x0 a. (-0, 0) b. (-0, 0] [-1, 1) d. (0, 1] e. [0, +0) с.

A: A function f(x) is said to be discontinuous at x=a if limx→a+f(x)=limx→a-f(x)=f(a)

Q: Let f(t) = 2t² for -4 < t <4 and extend periodically. Compute: f(1) = f(5) = f(-4.5) = f(-8)= f(12)…

A: Periodic function: A function is said to be periodic with period T if f(x+T)=f(x).

Q: Describe the interval(s) on which the function is continuous. (TX) f(x) = sec

A: f(x)=sec⁡(πx4)

Q: Let f(r) = r". For what values of r does the expression n=0 converge? For these values of r, write…

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Q: Determine whether -2, if z < -2 *, if - 22122 |8- z), if 2<z <3 |4 -r, if z 23 f(a) = is continuous…

A: givenf(a)=-2, ifx&lt;-2x5 , if-2≤x≤28-x , if2&lt;x&lt;34-x2 , ifx≥3one limit of this piecewise…

Q: Denote the range of a function f, as z varies over a domain D, by f(D). The function f is called…

A: Consider the given function: The objective is to determine whether the function fz=ezis bounded…