1. (a) Give an example of functions f and g defined on R such that both...
7. Give an example of a functionf 0,R that is discontinuous at every point of [0, 1 but such that |f is continuous on [O, 1]
7. Give an example of a functionf 0,R that is discontinuous at every point of [0, 1 but such that |f is continuous on [O, 1]
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...
We specific example of two functions that are defined by different rules for formulas) but that are equal as 2. Consider the functions /(x) = x and g(x) = Vr. Find: (a) A value for which these functions are equal. (b) A value for which these functions are not equal. 3. Let A = {1,2,3,4), B = {a,b,c,d,e), and C = {5, 6, 7, 8, 9, 10). Let S : A +B be defined via ((1.d).(2.b), (3, e), (4.a) Let...
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that
2. Let f R R and g R-R be functions that are continuous on1,1 and differentiable on (1,1). Suppose that f(-1-f(1) and 9(-1). Show that there exists c e (1,1) such that
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...
(1)Give an example of a function f : (0, 1) → R which is continuous, but such that there is no continuous function g : [0, 1] → R which agrees with f on (0, 1). (2)Suppose f : A (⊂ Rn) → R. Prove that if f is uniformly continuous then there is a unique continuous function g : B → R which agrees with f on A.(B is closure of A)
(1 point) Suppose f, g: R² + R2 are continuous functions, where g is surjective. Determine if the following sets are open, closed, neither, both or if it can't be determined. 1.9-1 (R) 2. (f • g)-+ ({(1, 2)}) 3. (f+9) (B(0; 1)) 4. (f+g)-1({(x, y) : x > 0}) 5.9-1 (B(0; 1))
S f(r)da= g(x)dz. Prove a,bsuch that (8) Suppose f and g are continuous functions on that there is ro e (a, b) such that f(ro) = g(xo). (9) Prove that if the function f is continuous on a, b, then there is c E [a, b such that f(x)dax a Ja f(e)
S f(r)da= g(x)dz. Prove a,bsuch that (8) Suppose f and g are continuous functions on that there is ro e (a, b) such that f(ro) = g(xo). (9)...
Composition of two functions: Domai... Two functions g and f are defined in the figure below. 8 0 2 4 3 9 -9 Domain of 8 Range of g Find the domain and range of the composition fog. Write your answers in set notation. Domain of s Range of s Domain of fog : 0 Range of f •8: X ? Explanation Check 2020 M