|f(x) – f(xo)| < e whenever |x – xo] < 8. And the function f : R → R is continuous if it is continuous at every xo E R. Show that this definition of continuity of functions over R agrees with the topological definition of continuous functions.

Q: (x+2, x ≤ 1 (B)Test the continuity of the function f(x) = {4-x, x>1, at x=1. 6

A:

Q: Definition A function f(x) is continuous from the right at a number a if... And f(x) is continuous…

A:

Q: Let X be Hausdorff and f : X → X be a continuous function. Show that {x : f (x) = x} is closed in X.

A: Let, .We show  is open and so F is closed.

Q: Sketch the graph of a function f such that f (x) ≥ 0 on [0, 1] and f (x) ≤ 0 on [1, 2], whose…

A: Consider the function Then and The graph of the function is shown below,

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 if f(x) and g(x) are continuous at a and c is a constant, then the function cf(x) + g(x)…

A:

Q: アー)。 The xy-plane above shows the graph of a continuous function f that is defined on interval [-2,…

A: We use the graph of f(x) and the definition of the function g(x) to answer the given question.

Q: Suppose a smooth function f(x) satisfies f(0) = 0, f(1) = 2, and that f[x0,x1,x2] = 1  for any three…

A:

Q: Show that the following relations are not functions on R. a) ƒ = {(x, y) = RxR | y=x/(x² −7x+12)}

A: Given function. y=xx2-7x+12 is not define for x2-7x+12=0 =&gt; (x-3)(x-4)=0

Q: Suppose f: IR→ R and g: R → R and h: R→ R are continuous functions such that h(x) = f(x)g(x) for…

A: Using given information we have to tell that whether the  given statement is true or false.

Q: Give an example of a function f : (X1, d1) → (X2, d2) that sends compact sets to compact sets, but…

A:

Q: The percentage of female high school graduates who enrolled in college is given in the table below.…

A: Let x represent time(in years) since 1970, and let y represent the corresponding percentage of…

Q: For real numbers m and b with m + 0, show that the map f: R-→ R defined by f(x) = mx + b is…

A: Given m and b be the real number with m≠0 and f:R→R defined by fx=mx+b. We have to show that fx is…

Q: Show, using the definition of continuity from the topological point of view, that the constant…

A: See the attachment.

Q: 3. Let z € R" be denoted by x = (₁, 2,...,n) and || x ||:= √√x² + x² + +x². Let f: RR be defined by…

A:

Q: Using the definition of continuity directly prove that f : R → R defined by f(x) := x2 is continuous

A: Using the definition of continuity, the given question has been proved in step 2.

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.