Prove: If a function is differentiable on an open set U, then it is continuous on U.
We will use epsilon-delta definition of differentiability and from there we prove the continuity of a function f.
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Prove: If a function is differentiable on an open set U, then it is continuous on...
Let U be an open subset of R". Let f:UCR"-R be differentiable at a E U. In this exercise you will prove that if ▽f(a) 0, then at the point a, the function f increases fastest in the direction of V f(a), and the maximum rate of increase is Vf(a)l (a) Prove that for each unit vector u e R" (b) Prove that if ▽/(a)メ0, and u = ▽f(a)/IV/(a) 11, then
Let U be an open subset of R". Let...
Let U be an open set of R3 and f:U Ri is a smooth function. Prove that the graph of f.e.i. the set M={(u,f(u)) E R4: ueU} is a 3-dimensional submanifold of R*.
Let U be an open subset of R. Let f: U C Rn → Rm. (a) Prove that f is continuously differentiable if and only if for each a є U, for each E > 0, there exists δ > 0 such that for each x E U, if IIx-all < δ, then llDf(x)-Df(a) ll < ε. (b) Let m n. Prove that if f is continuously differentiable, a E U, and Df (a) is invertible, then there exists δ...
Let frutiv be a continuous, complex-valued function on the connected open set U. Which of the following statements is equivalent to the analyticity of f on U? o af For every zo EU, there are coefficients ao, al, ... such that > an(z – zo)" n=0 converges for every z EU, Uz = Vy and vz = Wy == 0 for all rectangular paths y in U. None of these.
Prove the following Theorem:
Theorem. f : X → R is continuous + for any open set U C R, the pre-image f(U) is open in the domain of X (i.e., f(U) = XnV for some open set V C R).
Let f be defined on an open interval I containing a point a (1) Prove that if f is differentiable on I and f"(a) exists, then lim h-+0 (a 2 h2 (2) Prove that if f is continuous at a and there exist constants α and β such that the limit L := lim h2 exists, then f(a)-α and f'(a)-β. Does f"(a) exist and equal to 2L?
Let f be defined on an open interval I containing a point a...
*4, Let U be an open subset of R" and f:U-R" a function whose component functions have continuous partial derivatives. We say that f is an immersion if Dsf is injective for all v in U and a submersion if Dof is surjective for allv in U. (a) Suppose that f:U-R" is an immersion. Prove that, for each v in U, we can find an open set V of U containing v, an open set W of R" containing f...
At x = 1 the function (in the graph below) is o continuous, differentiable o continuous, not differentiable o not continuous, not differentiable o not continuous, differentiable
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem)
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
1. For differentiable vector functions u and v, prove: u'(t) X v(t)+ u(t) X v'(t) lu(t) X v(t)] 2. For the differentiable vector function u and real-valued function f, prove: lu(f(t)))= f(t)u' (f (t))
1. For differentiable vector functions u and v, prove: u'(t) X v(t)+ u(t) X v'(t) lu(t) X v(t)] 2. For the differentiable vector function u and real-valued function f, prove: lu(f(t)))= f(t)u' (f (t))