. Consider the function f: R→ [0, 1] defined by f(x) = 1 if x is rational, and f(x) = 0 if * is irrational. For which values of a € R does lima f(x) exist? At which values of a € R is f continuous?

Q: For what value of the constant c is the function f(x) continuous on the interval (-∞, ∞) where 4c -…

A: Give query is to find the value of c.

Q: 2. Consider the function / with domain (-2, o0): 3|2 – 1| 2 + 27 - 4 if – 2 <z<1 f(z) = if r = 1…

A:

Q: Which of the following functions are monotonic on their natural domains? Check all that apply. O y =…

A: We have to find the functions which are monotonic on their natural domains.

Q: Consider the function f defined by the formula [ Ae*-² + cx- 3c f(x) = x³ + cx² + B if x < 2, if x ≥…

A:

Q: For what values of a is Jaʼx – 2a, x2 2 f(x) ( 12, x< 2 continuous at every x?

A: Given : The function is f(x) =a2x - 2a        , x ≥212,                     x&lt;2 .

Q: don't understand how the equation turned into 5x+x+6/x(x+6), could you go through the equation again…

A:

Q: 2. Determine if a function f : [0 ; 2pie] → R defined by f(x) = sin 2pie is continuous

A: Determine if a function f : [0 ; 2pie] → R defined by f(x) = sin 2pie is continuous.

Q: 3. We study the function f defined below: { f(x) = (1) (2) if x is rational if x is irrational Prove…

A:

Q: Suppose that f : R → R and g : R → R are related by the formula f(x) = g(x)+(x−a)n. For which values…

A: Consider the given formula. fx=gx+x-an And, fx=gx+x-angx=fx-x-an

Q: D. Prove that f(x) = |x - 6| is continuous at all points c in R.

A: Given function is fx=x−6. We have to prove that the function fx=x−6 is continuous at all point c in…

Q: Suppose that f: R20 R20 is defined by f(x) = x². What can we say about f? →

A: Sol:- Given :- f:R≥0→R≥0It is clear that doamin of Function fx ≥0 and Range of function fx is also…

Q: A. Consider the function defined by [x² +1, if x<5 f (x)={ |6x– 4, if x25 a. Is f continuous at x =…

A:

Q: Given a surjective function ? ∶ [−1, 1] → [−1, 1], prove or disprove: ? −1 (? {0}) = {0}.

A:

Q: For what value of k is the function f continuous on (-o0, 00)? z – 1 ka² – a + 23 if a > -4 if z <…

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Q: For what values of a is Ja'x – 2a, x 2 2 f(x) : \ 12, x < 2 continuous at every x?

A: Given function is Since f(x) is continuous for every x. So, f(x) is continuous at x=2.

Q: 1. For each of the following statements, determine whether it is true or false and justify your…

A:

Q: Question 6 Prior to running the Boston Marathon, a runner's personal best pace for a race is 10…

A: Given that Prior to running the Boston Marathon, a runner’s personal best pace for a race is 10…

Q: Consider the function g defined by x x x + 4 Is continuous at x = = 3? а. g b. Is g continuous at x…

A: Note that, according to Bartleby guidelines, we can answer only the first 3 sub-parts in the case of…

Q: (a) Let x2 if x is rational f (x) = 0 otherwise. Prove that f is differentiable at 0.

A:

Q: Find the absolute extrema, if any, of the following functions on the given interval: a. f(x) = on [–…

A: Given

Q: 5. Consider the function g(x)=1 a) For what values of x is g(x) continuous? b) For what values of x…

A: In the question we have to find where function is continuous and differentiable.

Q: Which of the following functions on R is sublinear? a). s1(x) = # x if x ≥ 0 0 if x 0

A: Definition: Let X be a vector space. A function p: X→ℝ is said to be a sublinear function, if it…

Q: c. Let f be differentiable on R. Suppose that f'(x) # 0 for every r. Prove that f has at most one…

A: We can explain this statement by 2 explainations given below

Q: 19. The function f is defined by f:x 1+x*ER + f is injective. B f is surjective. C f is neither…

A: Please see the explanation below.

Q: d. For each of the following, state whether the functions f: R → R, x → f(x) are injective,…

A: Given functions: fx=3x+2 fx=3x3-7x+3 fx=x6 fx=sinx fx=0.01x fx=4 We have to check whether the…

Q: Prove or disprove: f : Z30 → Z15 given by f([x]) = [x²] is a well-defined function.

A:

Q: Let f(x) = x³ – 6, – 3 < x < - 1 The domain of f- is the interval [A, B 1 where A and B = %3D

A:

Q: c) Consider a function f defined on the real numbers as { 0ifais irrational. S(r) = 1/n ifr= m/n…

A: To show that the function f is continuous at all irrational numbers.