Let f: [1,1]→ R be a continuous function. Prove the following statements: (a) If there is c = [-1, 1] such that f(c)f(−c) < 0, then there is de R such that f(d) = 0. (b) If ƒ([−1, 1]) = (-1, 1), then f is not continuous.

Q: Let f : R → R be a function with the following property: |f (a) – f(b)| < 4|a – b| for all a, b E R.

A:

Q: В. B. Show that the function f (z) = -i z2 is analytic.

A: The solution is given as

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

A:

Q: Suppose f is differentiable on the set of real numbers R and a E R. Let F(x)= f(xª) and G(x) =…

A: F(x)=f(xα)F'(x)=f'(xα)ddx(xα) , [chain…

Q: Let R={(x,y) ERxR: |x| = |v|). Then R a function on R. оо True False

A:

Q: Q3-A) Is the function F(Z) continuous or not at z-U? 1+|21,

A: Let z = x+iy then Re(z) = x and |z| = √(x^2 + y^2) If z = 0 =&gt; x = 0 and y = 0

Q: 5. Prove that the function f: Z → Z defined by f(x) = 3x + 1 is one-to-one.

A:

Q: If y =f (x) is a function in x then prove that f(a + nh) ="C,f(a) +"C, Aƒ(a) +"C,ƒ(@) + +"C, A"…

A: Given: y=fx is a function in x Prove that: f(a+nh)=C0nfa+C1n∆fa+C2n∆2fa+…+Cnn∆nfa…

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.

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…

A:

Q: the following statement. 'If yes, explain why. If no, provide a counterexample. For every real…

A: If the function is injective , then the above statement is true.

Q: Let f be a real-valued function continuous at the point a = (a1, a2) in R². Keep a2 fixed and define…

A: Given that f is real valued function and continuous at a=a1,a2 in ℝ2 and gx=fx,a2. To show : g is…

Q: Suppose that the function f: [0, 1]  in R is continuous and thatf (x)&gt;= 2     if 0 &lt;= x &lt;…

A:

Q: Assume ƒ is a continuous on [2, 4],, and ƒ(2)= 4. Evaluate ƒ(4).

A: we have been given that f is continuous on [2,4] and f(2)=4. we have to evaluate f(4). the given…

Q: Let f: R2 R2 be the function given by f(x, y) = (x + y, y). Prove or →>> 8. disprove: f is…

A: Function which is one to one

Q: (c) Evaluate the Riemann integral ₁¹(x) dx by using definition with the parti- tion 2n=1,2}, n+1 =…

A:

Q: Let f(x) = x. Show directly that f ∊ R[0, 2]

A:

Q: Use the three-steps definition of continuity to show that the function f(z) = z² - 2z+ (1 + i) is…

A:

Q: 1. Explain what it intuitively means for function f(x,y) to be continuous in some domain D.

A: To explain the meaning of the given statement.

Q: Consider a function f given by А f(x, y) u(x) + iv(y). Determine all possible functions f of the…

A:

Q: f continuous at 3

A: We have to check that f is continuous at 3 or not. It is given that f(q)=3q for q rational number…

Q: Prove that the function f : R → R given by f(x) = 3x° – 2x is continuous at 1 using the a.…

A: