15. Let f. g : R → R be continuous at a point c, and let h(x) := sup (f (x). g(x)} for x e R. Show that h(x) = (f(x) + g(x)) + I (x) - g(x)| for all x e R. Use this to show that h is %3D continuous at c.

Q: 1. Show that the function, defined for x E R by f(x) = 1 is uniformly continuous 1+x² on R.

A:

Q: If f is continuous but not differentiable at x = c then there is a corner on its graph at (c, f(c)).…

A:

Q: Is f(x)= Vx + 1 continuous at x == -1

A: Draw the graph and check continuty with help of limit of function.

Q: For the continuous function f(x), if f’(-4) does not exist, then x=-4 is a critical point.

A:

Q: #46

A: We need to find the value of b such that the function g(x) is continuous at every x,

Q: Consider g(x) = x!/5. Is g(x) differentiable at x=0? Is g(x) continuous at x=0?

A: Given function

Q: 17. Show that f(x) = x² +2x is uniformly continuous on the interval [0, 10], using the e -8…

A:

Q: Observe that f (x) = x 2 is continuous on [0, 1] with f(0) = 0 and f(l) = 1. Because f (0) < 0.5…

A: f(x)=x2 is continuous on [0,1] with f(0) = 0 and f(l) = 1 we need to prove that according to…

Q: Let f(x) be a continuous function on R-{0} such that (f(x) dx exists for all a > 0.If g(x)= |- dt, t…

A:

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

A:

Q: Suppose f(x) < g(x) and both are continuous on a, oo). If / g(x) dx converges, then so does f(z) dz.

A: Since it is improper integral so we will use the limit concept to prove these fact.

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: Prove that f(z)=|z| is nowhere differentiable in C but is continuous everywhere.

A: Given function is fz=z. We have to prove that the function fz is nowhere differentiable in C but is…

Q: Px (w) is a continuous function of w in (- ∞, 0) for any real value of ow, and in fact, it is…

A:

Q: c. Show that f(x) is NOT uniformly continuous on [0, +∞).

A:

Q: Suppose that f(1) = 1, f(4) = 6, f (1) = 7, f '(4) = 5, and f " is continuous. Find the value of xf…

A:

Q: 2. Show that the function, defined by f(x) = is NOT uniformly continuous on its x2 %3D domain D(f) =…

A:

Q: 2. If f is a continuous function such that S(t)dt = cos(2x)e* - [e s(t)dt for all x, find an…

A:

Q: Show that f(x) = In x is not uniformly continuous on (0,2)

A:

Q: Suppose that  ff"' is continuous and f'(c)  =f"(c) =0 , but f"'(c) &gt;0. Does f have a local…

A:

Q: (b). Show that iff= u+iv is analytic in a region S and u is a constant function (i.e., independent…

A:

Q: It is frequently required to solve equations of the form f(x) = 0. When f is a continuous function…

A: To solve an equation fx=0, if f is continuous on an interval a,b such that fa and fb are of opposite…

Q: Given a continuous function of which is twice differentiable, if f" (c) > 0, which of the following…

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: Suppose that f(1)=8, f(4)=−4, f′(1)=−3, f′(4)=2, and f′′ is continuous. Find the value of ∫from 1 to…

A:

Q: how (in terms of  − δ) that a function f : R 3 → R defined by f(x, y, z) = (2x + 3y + 4z) is…

A: Solution of given problem is followed by

Q: Is g(x) = ln( x2+2 ) uniformly continuous on the interval (−∞,∞)?

A: given function is                                                        g(x)=lnx2+2 log is defined…

Q: 1. Show (in terms of  ε - δ) that a function f : R3 → R defined by f(x; y; z) = (2x + 3y + 4z) is…

A:

Q: that makes y continuous but not differentiable by writing the required procedures to that If: X≥0…

A:

Q: 2. Show that the function f(x) = cos x is uniformly continuous on

A:

Q: 4. Give an example of a continuous function f : (0, 1) → R that cannot be uniformly approximated by…

A:

Q: Suppose that a continuous function g satisfies: | = 36 and g(x)dx | 9(2x)dr = 15. 0. : = =p(= |…

A:

Q: 4. If L is a Continuous function on H, then unique YEH such that L(x)=(x,y). уєн there is a

A: Given that L is continuous on H. To show: y is unique in H.

Q: Find a value of k making h(x) continuous on [0, 5]. kx , 0 < x < 1 h(x) = x +3, 1<x < 5

A: Given: hx=kx,0≤x&lt;1x+3,1≤x≤5 given function will be continuous over 0,5, if it will be continuous…

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: 50 if x E Q 1 if x € R\ Q Let f(x)= Prove that y=f(x) is not continuous at any points a E IR.

A: Take p ∈ Q and let (xn) be a sequence of irrationals converging to p. Then f (p) = 1 but f (xn)→ 0…

Q: 1. Suppose f(r) is a continuous function on r E [a, b]. Prove f(x)dx < Slf(x)| dx.

A: According to our guidelines we can solve only first question and rest can be reposted.Thank you

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: Prove that if m(x) is differentiable on (−∞,∞) and its derivative m'(x) is bounded then m is…

A:

Q: Suppose that f is a continuous function on R and, g and h are differentiable functions on R.…

A:

Q: If f(z) is continuous at z then it must be differenttiable at z, 0.

A:

Q: (3x² -1, x 1, Determine c and d so that f(x) is continuous everywhere.

A:

Q: Show that x³, if x 0 is not continuous at x = 0. %3D

A: According to the problem, we have

Q: 1. Estimate the limit of the signum function s(x) as x approaches zero from the right. (-1, x0

A:

Q: 3. Consider the function g(x) defined by { x² sin(1/x) if x > 0, g(x) = if x <0. (a) Show that g(x)…

A:

Q: Suppose that f (x) is a continuous function. 3 If | f(x) dx=5, then what is the value of z - f(x)]…

A: Given, ∫13f(x) dx =5Now,∫13[x -f(x)] dx= ∫13x dx -∫13f(x) dx =x2213 -5…

Q: Σ 1 is continuous function in R. Prove f(x) = 4 n=1 n' +x*

A: The given function is                                     f(x)=∑n=1∞1n4+x4 To show that the above…

Q: Suppose that f(x) is a continuous function on the interval [0, T] . If f(x)=(sin x)ª for 0<x<T, then…

A: we have to find the value of given function. the solution is following.

Q: 1. Use the e-ổ definition to prove that f(x) = x² is uniformly continuous on [a, 이.

A: