Let f:R→R satisfy f(x+y) =f(x) +f(y), ∀x,y∈R. If limx→0f(x) exists, prove that f is continuous on R

Q: Let f: A→ X X Y be defined as f (a) = (f₁ (a). f2 (a)) each a ≤ A, Then f is continuous if and only…

A:

Q: Define f : R2 → R by letting (y² – x)² (y4 + x²) f(x, y) = if (x, y) # (0, 0), and f(0,0) = 1. ||…

A: The given function is f:ℝ2→ℝ defined as: fx,y=y2-x2y4+x2if x,y≠0,01if x,y=0,0 Claim: The given…

Q: Let f: R³ R be the function defined as f(x, y, z)= √√₂² + y² + 2², 0. if (x, y, z) = Q³, otherwise.…

A:

Q: Let f [0, 2] → R be a continuous function, with ƒ(0) ≤ f(1) ≤ f(2). Show that there : exists xo [0,…

A:

Q: Let f: X→Y be a function, and let A,BCY. Prove or disprove: f-¹(A)uf-¹(B).

A: The given statement : f-1(A ∪ B) = f-1(A) ∪ f-1(B) is true and we shall prove this statement.

Q: Determine all functions f : R → R that have the property that x + y f (x) – f(y) "() - 1e) 2 x – Y…

A: Since x and y are arbitrary points  and f'exists. hence F is continuous and also differentiable. let…

Q: Show that the function f defined by (x, y) = (1, –1) f(x, y) = r? + y (x, y) # (1, –1) x + y is not…

A:

Q: Let f: R (-1, 1) be a function defined by Prove that (a) f is injective; (b) f is surjective. f(x) =…

A:

Q: → R be a function defined by 3x² + x+5 f(x) = not continuous at x = = 0.

A:

Q: Let f : Z+ → N be the floor function f(x) = [(x – 1)/2].

A:

Q: Consider the function f: RR defined by 1 (a) (b) By using definition, explain whether f is…

A:

Q: Let f : R → R be defined by f(x) = x^2. Is f one-to-one? Explain your answer.  Is f onto? Explain…

A:

Q: Let ƒ : R\{1} →→ R\{1} be defined by f(x) = x/(x - 1). Show that ƒ is bijective and find f-¹. Also,…

A:

Q: Show that f: R\{2} → R\{5} defined by f(x) = (5x+1)/(x - 2) is bijective.

A:

Q: Let /: X →Y be a function, A, BC X and C, DCY. Show that: 1. f(AUB) = f(A) uf(B) 2. If AC B, then…

A:

Q: Let h : R → R be some twice-differentiable function. Define f(x,y) = h(x) for all x ≠ 0. Show that…

A:

Q: What does is mean for two functions f(x) and g(x) to be orthogonal

A:

Q: Give a pair of functions f: R →R and g : R →R such that g is not tangent to f at 0 and yet lim,0…

A: To give a pair of function f : ℝ→ℝ and g : ℝ→ℝ such that g is not tangent to f at 0 and yet limx→0…

Q: f: R2 → R, f(x1, x2) = x1-x2 function is given. a) Can f be an surjective function? b) Can f be an…

A: f: R2 → R, f(x1, x2) = x1-x2 function is given. a) Can f be an surjective function? b) Can f be an…

Q: If f is differentiable at xo, prove that f (xo + h) – f (xo – h) – f' (x0) - lim h-0 2h

A: Given that: f is differentiable at xo   To prove: limh→0f(x0+h)-f(x0-h)2h=f'(x0)

Q: If f is a continuous function satisfying f(x) = x* - x+(1-x)[,f'f)dt + x[ (t -r°)f(@dt for all x,…

A:

Q: d) Let f be a continuous function. Then 1 [ S f(x,y) dady = f*f* f(x,y) dyda. 0

A:

Q: Let f be a differentiable function such that f(h) f(x + y) = f(x) + f(y) + 2xy and lim = 15. Find…

A:

Q: If fis a bijection, find the sets X and Y.

A: