h(x) = 3 = x + + n=1

Q: if x 4.

A: The function gx is a piecewise defined function.

Q: Suppose f: (0, 1)→ (0, 1) is a surjective monotonic function. True or false: The function f is…

A: f: 0,1→0,1 is a surjective monotonic function.

Q: Suppose that f is a continuous function on [0,2] such that f(0) = f(2). Show that there is a real…

A:

Q: 2. Suppose that f is an increasing function on the interval [a, b]. Let F(t) = %3D a.) Argue that if…

A: suppose that f is an increasing function on the interva a,b let Ft=∫0tfxdx (a) if a<x*<b then…

Q: 6. Let f be continuous on [a, b]. Show ffdF = f(x) [F(b) – F(a)] for some x € [a, b].

A: This is a consequence of the intermediate value theorem

Q: True or false: If f: (0, 1)→ R is defined by f(x)=√x for every real number a in the open interval…

A: fx=x for all real xin the open interval  0,1

Q: Prove that there is an additive function T : R --> R that is not linear.

A: Let T: V→W  be the linear transformation and we it is linear,…

Q: [0,1] 6. If f is continuous on • prove that

A: We can solve an integral of some function f(x) w.r.t. x by using substituting, say, t=f(x) which…

Q: Prove that x² +¸µ111² +21112 where prime denotes differentiation w.r.t.s. = 1 o² + 1+p¹² P

A:

Q: 5. Let f be defined for all real numbers and suppose f(x)-f(y)| ≤ (x - y)² for all real numbers, x…

A:

Q: Determine b so that f(x) is continuous if O 8 x + 6 xS 6 f( x ) = D O 2 x2 + bx +10 x > 6 b =

A:

Q: Show that f is continuous on (∞ , -∞ )  .

A: we have to show that f(X) is continuous in (-∞,∞). f(x)=x2             if   x<1           x…

Q: 4. Let f be defined for all real x, and suppose that \fx) – f(v)| < (x – y)? for all x, y e R. Prove…

A:

Q: 6. Prove that |xy| = |x|ly| for all real numbers x, y.

A: The given problem is to prove the following statement. Given statement is |xy|=|x||y| for all real…

Q: Consider the PIECE-WISE function: Ат-В T-2 if r<1, f(x) = { 32, if 1<x < 2, Ba? — А, if a 22.

A:

Q: Show that the function Arg(z) is not continuous on each point of the non-positive real axis.

A:

Q: Define f: R -R by Sir +1 if r 4. Find all real values at which f(x) is continuous. Prove that your…

A:

Q: f(c) = g(c).

A:

Q: Show that x + is increasing on [1, 0).

A:

Q: Which of the following interval is f(x) = O (2, +∞0) O [2, +∞0) 0 (-∞0, +∞0) ○ (0, +∞) 1 Vx…

A:

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

A:

Q: Let A =[1 -1]            [2 0] and --->x =[ i]                  [1 + i] . SHOW that Re (A--->x )…

A:

Q: t 0, if x is irrational 1, if x is rational ve a rigorous proof that f is not integrable on the…

A:

Q: If f is continuous on the interval [5, 12] and f(5) = 7 and f(12) = -2, then f must have at least…

A:

Q: Find the F.T. of the fall wing Signals (functions): F(t)=1+cos(6πt +)

A:

Q: Let f (x) = x2 if x is rational and f (x) = 0 if x is irrational. a) Prove that f is continuous at…

A:

Q: Suppose the function ƒ has the property that IS(x) – S()| < |x -|| for each pair of points x, t in…

A:

Q: Would it be possible for a function to be continuous on [0,2) but not on [0,2]?

A: Let us consider the function f(x) =(x+4)/(x-2) .

Q: 4. `s.] Find the number a and b at which f is continuous everywhere? x² -1 if x 2

A:

Q: Show that the function f:R to R defined by f(x)=5x-12 is continuous using the E-d definition

A:

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

A:

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.