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Let f(x) be a continous function defined on R. Consider the following function, g(x) = max{f(t)\t...
2. Let f:R + R and g: R + R be functions both continuous at a point ceR. (a) Using the e-8 definition of continuity, prove that the function f g defined by (f.g)(x) = f(x) g(x) is continuous at c. (b) Using the characterization of continuity by sequences and related theorems, prove that the function fºg defined by (f.g)(x) = f(x) · g(x) is continuous at c. (Hint for (a): try to use the same trick we used to...
In this problem we consider only functions defined on the real numbers R. A function f is close to a function g if 3x E R s.t. Vy E R, A function f visits a function g when Vz E R, R s.t. a<y and f() -g) For a given function f and n E N, let us denote by n the following function: n(x)-f(x)+2" Below are three claims. Which ones are true and which ones are false? If a...
Let f : [0,∞) → R be the function defined by f ( x ) = 2 ⌊ x ⌋ − x? where x? = x − ⌊x⌋ is the decimal part of x. Prove that f is injective. Let f: 0,00) + R be the function defined by f(3) = 212) where ã = x — [x] is the decimal part of x. Prove that f is injective.
3. In this problem we consider only functions defined on the real numbers R A function f is close to a function g if r e Rs.t. Vy E R, A function f visits a function g when Vr E R, 3y E R s.t. For a given function f and n E N, let us denote by fn the following function: Below are three claims. Which ones are true and which ones are false? If a claim is true,...
Theorem 10.1.15 (Chain rule). Let X, Y be subsets of R, let xo e X be a limit point of X, and let yo e Y be a limit point of Y. Let f : X+Y be a function such that f(xo) = yo, and such that f is differentiable at Xo. Suppose that g:Y + R is a function which is differentiable at yo. Then the function gof:X + R is differentiable at xo, and .. (gºf)'(xo) = g'(yo)...
1. Let f:R → R be the function defined as: 32 0 if x is rational if x is irrational Prove that lim -70 f(x) = 0. Prove that limc f(x) does not exist for every real number c + 0. 2. Let f:R + R be a continuous function such that f(0) = 0 and f(2) = 0. Prove that there exists a real number c such that f(c+1) = f(c). 3 Let f. (a,b) R be a function...
In this problem we consider only functions defined on the real numbers R. A function f is close to a function g if 3r E R s.t. Vy R, A function f visits a function g when Vz E R,3y E R s.t. < y and lf(y)-g(y)| < We were unable to transcribe this imageBelow are three claims. Which ones are true and which ones are false? If a claim is true, prove it. If a claim is false, show...
(b) Let f 0, 1-R be a C2 function and let g, h: [0, 00)-R be C1. Consider the initial-boundary value problem kwr w(r, 0) f(a) w(0, t) g(t) w(1, t) h(t) for a function w: [0,1 x [0, 0)- R such that w, wn, and wa exist and are continuous. Show that the solution to this problem is unique, that is, if w1 and w2 [0, 1] x [0, 00)- R both satisfy these conditions, then w1 = w2....
real analysis II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest terms. 1. Prove that f is discontinuous at every x E Qn [0,1]. 2. Prove that f is continuous at every x e [0,1] \ Q. II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest...
Please write neatly Let f(x) be a Cl(R) function and u(t, x) be a solution to the inviscid Burger's IVP Ut + uuz = 0, 4(0,x) = f(x). (a) Use the formula ult, x) = f (x – ut) to derive the following explicit identity for the partial derivative Ou du Ət f($)f'(5) 1+tf'(E) with $ = X – ut. (b) Assume that there exists a number wo E R such that f'(20) < 0 and f(x) > 0. Use...