Let f : R → R be a function and M E R, such that for every (x, y) E R², the function g(t):= f(xt, yt) is differentiable and |g'(1)| < M. (a) Show that f is continuous at (0,0). (b) Find an example of such an f that is discontinuous at every other point of R2. Hint: Think back to how we constructed a nowhere continuous function on [0, 1].

Q: Assume that is differentiable on a ≤ x ≤ b and that f (b) < f (a). Show that f' is negative at some…

A:

Q: Show that the function f (z) = Z is continuous at the point z=0 but not differentiable at z=0.

A:

Q: Theorem 11. Let f be a continuous function on R such that f(x+y)-f(x)+fv) vx, yER. Show that f(x)3cx…

A:

Q: Let f be a differentiable function on R. Prove that between any two roots of f(x) = 0, there is at…

A: We have to prove that between any two roots of f(x) = 0, there is at least one root of f(x) + f'(x)…

Q: x² - y² z = g(x, y) = x² + y² 0 for for (x, y) = (0,0) (x, y) = (0,0)

A: Given function is z=gx,y=x5−y5x4+y4 for x,y≠0,00            for x,y=0,0. We have to check whether…

Q: b) Give an example of a function g: R →R that is continuous at some xE R, but not differentiable at…

A:

Q: Where is the function f(r, y) = x/y continuous?

A:

Q: Assume that f and g are differentiable functions such that f(0) 2 g(0) and that for all r€R, f'(z) >…

A:

Q: A) If g'(x)>0 for all x in the interval (4,7) then g(x) is for all x on the interval (4,7).| B) If…

A:

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

A:

Q: (b) Consider the function f : R → R, f(x) = { (i) Prove that f is discontinuous at 0. (ii) Prove…

A:

Q: 3. Let A be the set of all functions ƒ : 0, 1] → R that are continuous in the interval [0, 1].…

A:

Q: Define S = Qn [0, 1] and the function f: SS that satisfies f(0) = 0 and when p and q are 1 coprime:…

A:

Q: Prove and show complete solution: Let ƒ be defined on an open interval (a, b) and assume that ƒ has…

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: |Consider the function f with domain (-2, 0): 3|1² – 1| 21² + 2r – 4 if – 2<1<1 f(z) = if x = 1 3 if…

A:

Q: Let f: IR be a function. (a) Prove that if f is differentiable at some c I for which f(c) ‡ 0, then…

A:

Q: Determine the set of points domains at which the function is continuous.  G(x,y)= 2x tan y

A: To Find: Set of points in domain at which the function 2x tany is continuous.

Q: Assume that f and g are differentiable functions such that f (0) > g(0) and that for all x E R,…

A:

Q: be de by 8 <x3 + y3 f(x, y) = x2 + y2 if (x, y) = (0, 0) if (x, y) = (0, 0). %3D (a) Is f continuous…

A: Note- Since you have posted a question with multiple sub-parts, we will solve first three sub-parts…

Q: C is the curve y = e® from (2, e²) to (0,1) and F = (x², -y), compute ſ, F-dr.

A: Given that, C is the curve y=ex from 2,e2 to 0,1 and F=x2,-y Then, F=x2,-y=x2,-ex So, F→=x2i^-exj^