If a function f : R” → R is differentiable at Xo, prove that ƒ is continuous there. Let ƒ : R" → R. Assume that there is an affine function T(X) such that T(Xo) = f(Xo) and f(X) - T(X) lim X-Xo ||X-Xo|| ² Must f be differentiable at Xo? Prove or give a counterexample. = 0.

Q: Consider a function f : D → R, where D C R. Suppose the function satisfies the property f(xy) = f(x)…

A: A function f:D→R, where D⊂R f(xy)=f(x)+f(y) for all x, y in D f(x) is not zero for all x in D.

Q: Suppose that the function f : R → R is continuous for x ≥ 0, differentiable for x > 0, and satisfies…

A: Mean Value Theorem Suppose  f ( x )  is a function that satisfies both of the following conditions.…

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: If f is differentiable on R and f(0) = 1,|f0(x)| < 1,∀x ∈ R. Show that |f(x)|≤|x|+ 1,∀x ∈ R.

A: Note that |f0(x)| &lt; 1 should have been |f'(x)| ≤ 1. Otherwise, the given inequality won't be…

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: Let f: A → B be a function and let a ∈ A and b ∈ B. Does this mean it is true that f(a) = b? If not,…

A: We use definition of function .

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

A:

Q: 1. v₁ = (4, −2, 6, −4), V₂ = (6, −3, 9, −6)

A:

Q: Use the e-6 definition of continuity to prove that the following functions are continuous at the…

A:

Q: Let f [0, 1] → [0, 1] be continuous. Show that there exists an element x = [0, 1] such that f(x) =…

A:

Q: Suppose f and g are differentiable functions on an interval I. Show that if f′(x) = g′(x) for all x…

A: It is given that, f′(x) = g′(x) for all x ∈ I and f(x0) = g(x0) for some x0 ∈ I.