: Iff (z) is analytic at zo. (a) prove that it must be continuous at zo.

Q: If f is differentiable at a, then f is continuous at a.

A: If a function fx is differentiable at x=a then function must be continuous at x=a. But a function…

Q: Let h: R → R be a continuous function on all of R. If h(x) Tt, prove that h(t) 50.

A:

Q: Answer true or false. If f is not piecewise continuous on [0, co>), then L{f(t)} will not exist.…

A: The given statement is

Q: 3. If f : R R is continuous at ro E R and f(ro) > 0, prove that there exists an interval containing…

A:

Q: Use the limit definition of definite integral in the form of f(æ) dæ = lim {(#:)Ax, i=1 6 - a and r;…

A:

Q: We used the e-d characterization of continuity to show in class S (x) = x² is continuous at r = 3.…

A: Given: To explain the given statement as follows, is continuos at x=3.

Q: f(x+y)=f(x)+f(y) and f is continuous at 0. Prove that f is continuous at any a.

A: To show that the continuity at x=0 implies continuity at all x (under the given condition on the…

Q: Consider the piecewise function 2+x² x<0 x=0 g(x) = 1, 2-³, 0<x< 1 2-x x 21 (a) For each of the…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Let F(x) –3 cos(ť²) dt. Use the Fundamental Theorem of Calculus to find F'(x). F'(z) =

A: Given: F(x)=∫x−1(−3cos(t2))dtF'(x)=ddx∫x−1(−3cos(t2))dt=?

Q: For what value of b is X - b b + l' x 0 continuous at every x?

A: The given function is gx=x-bb+1if x&lt;0x2+b if x&gt;0. If the above function is continuous at x=0,…

Q: Use the limit definition of the definite integral, lim f(r,)Ar = (x) dr, %3D to prove that S(x) dr -…

A:

Q: Find a and b such that f(x) is continuous at x=1. [x – 2] + a x 1

A: A detailed solution is given below.

Q: Let h: R → R be a continuous function on all of R. If h(x) T, prove that h(T)<0.

A:

Q: Suppose h' is continuous x>0. IF x = 0 and h'(x) = 3√tan (2x)-1 it exists. i at Find h (0), 'F

A: For the solution of the problem follow the next step.

Q: If f(z) is continuous at z, then it must be differenttiable at z, ylgn O İhi O

A: If a function f(z) is continuous at z0 , then it may or may not be differentiable at z0. Hence, The…

Q: f(z) = Im z is continuous

A: Let us use the convergence criterion to prove that Im z is continuous. Let us assume that  znn=1∞ be…

Q: The function f is a composite of two continuous functions: y = cos x and y = x 2, so f is continuous…

A: The function fx is a composition of functions y=cosx and y=x2and it is a continuous function.

Q: Use Theorem 9.2.1 to show that T is not continuous at 0, no matter what value is chosen for b.

A:

Q: Prove that if f (x) is continuous at x = a and g(y) is continuous at y = b, then F(x,y) = f(x)g(y)…

A:

Q: If f is continuous everywhere except at x = 2, then f(x +1) dx is not integrable. True False

A: To check whether the given statement is True or False."If f is continuous everywhere except at x=2,…

Q: Define f : R → R by if a 1. Use the e-d characterization of continuity to show that f is continuous…

A:

Q: Suppose that f is continuous at 0, and that f satisfies f(x + y) = f(x)+ f(y) for all real numbers…

A:

Q: f is continuous at a if and only if |f| is continuous at a.

A: False

Q: Let h: R → R be a continuous function on all of R. If h(x) T , prove that h(T)<0.

A: Given that, let h:ℝ→ℝ be a continuous function on all of ℝ. Also given, hx≤0 for all x&gt;π.

Q: Prove that f (z) =zz is nowhere analytic

A:

Q: 0. If y = Ax) is continuous at x = xo, and z = yo =Axo), prove that z = g[f{x)] is continuous at…

A: Explanation of the answer is as follows

Q: Let f (z) = u + iv be an analytic function. If f'(z) = 0, prove using Cauchy-Riemann equations that…

A:

Q: If f is continuous on [0, 1] such that x* f(x) dx=0, for n0, 1, 2, 3,.. then show that f(x)=0 on [0,…

A:

Q: 2. Determine if a function f : [0, 27] → R defined by f(x) = sin 2n is continuous.

A:

Q: Show that the subtangent of f (x) = ex is everywhere equal to 1.

A: Formula to find the subtangent of the function helps to find the required subtangent. Derivative of…

Q: Suppose f> 0 is integrable on [a, b], continuous at xo € [a, b], and f(xo) > 0. Prove that f>0.

A: Given f≥0 is integral on a,b continuous at x0∈a,b and fx0&gt;0

Q: Prove that f(2) = z/(z* + 1) is continuous at all points inside and on the unit circle Iz] = 1…

A: we need to find those points where the denominator ( z^4+1) equals to 0

Q: Define f(0,0) in a way that extends f to be continuous at the origin. 6x² -x²y² + 6y² x² +y? f(x,y)…

A:

Q: Suppose f(z) is analytic for |z| &lt;= 1, and |f(z)| &gt;=1. If f(0) =1, prove that f(z) is…

A: To prove that the given conditions on the analytic function force it to be the constant function…

Q: show that f,given by f(x) x +0 )is continuous on R-{0}

A:

Q: At which point c does g have a removable discontinuity? How shouldg(c) be redefined to make g…

A: Removable discontinuity- if right limit gc+0 is equal to left limit gc-0 is not equal to gc, thengx…

Q: Consider g(x) = rº/3. Is g(x) differentiable at x=0? Is g(x) continuous at x=0?

A: We find g'(x) using the power rule  Then we find the left and right-hand limits of g'(x) and g'(0)…

Q: Let f(x) = 0 if x∈Q if x and f(x) = 1 if x ∈R\Q. Prove that y=f(x) is not continuous at any point…

A:

Q: Use the comparison theorem with f(x) = = 1 √x r∞ 1 •√₂ √I-1 to show that is divergent.

A: Solution is given below:

Q: Let g be a continuous function. Using the substitution u = 3x – 1, the integral (g (3x – 1)) dx is…

A: We can use the concept of method of substitution in order to solve the given problem.

Q: Let A := (0, 1] and let ƒ : A → R be defined by f(x) = !. Prove that ƒ is continuous on A.

A:

Q: If f is differentiable at a ,then f is continuous at a

A: Every differentiable function is continuous.

Q: Use the definition of continuity to show that h(x) = x^2 + 5x/2x + 1 is continuous at the number a…

A: The given function,   h(x)=x2+5x2x+1

Q: If f is analytic and has a pole at z = a, then f(2)|| → 0 as z → a in any manner.

A:

Q: Suppose that a continuous map g : U → W is tangent at a to a map f:U → W. Prove that f is also…

A: given that continuous map g:U→W is tangent at  a  to a map  f:U→W. we have to show that f is also…

Q: Prove f (x) 1+ mx is continuous at x = 1, where m e R

A:

Q: 3. Let f(r) = r sin() if z #0, and f(0) = 0. Show that f is continuous at r= 0 by using e- d…

A:

Q: Consider the following theorem. Theorem If f is integrable on [a, b], then FX) dx = lim ° fx;,)ax n-…

A:

Q: Let f,g: D -> R be conitnuous at c ∈ D. Prove that fg is continuous at c.

A:

Power Operation

Power operation is topic of algebra in Math. It is use to represent repeated multiplication. Very big number and very small number can be easily express using power operation. Power operation is useful in many fields. In space engineering, it helps in representing the distance or size of particular heavenly body. In medical field, it is used to represent very small size. In medical field it helps to mention size of bacteria or virus. 

Exponents

The exponent or power or index of a variable/number is the number of times that variable/number is multiplied by itself.