Reference no: EM132153089

Exercises -

Q1. Suppose f is a real function defined R¹ which satisfied

lim_h→0[f(x+h) - f(x-h)] = 0

Q2. If f is a real continuous function defined on a closed set E ⊂ R¹, prove that there exist continuous real functions g on R¹ such that g(x) = f(x) for all x ∈ E. (Such functions g are called continuous extensions off from E to R¹.) Show that the result becomes false if the word "closed" is omitted. Extend the result to vector-valued functions. Hint: Let the graph of g be a straight line on each of the segments which constitute the complement of E. The result remains true if R¹ is replaced by any metric space, but the proof is not so simple.

Q3. If f is defined on E, the graph off is the set of points (x, f(x)), for x ∈ E. In particular, if E is a set of real numbers, and f is real-valued, the graph off is a subset of the plane.

Suppose E is compact, and prove that f is continuous on E if and only if its graph is compact.

Q4. If E ⊂ X and if f is a function defined on X, the restriction of f to E is the function g whose domain of definition is E, such that g(p) = f(p) for p ∈ E. Define f and g on R² by: f(0, 0) = g(0, 0) = 0, f(x, y) = xy²/(x² + y⁴), g(x, y) = xy²/(x² + y⁶) if (x, y) ≠ (0, 0). Prove that f is bounded on R², that g is unbounded in every neighborhood of (0, 0), and that f is not continuous at (0, 0); nevertheless, the restrictions of both f and g to every straight line in R² are continuous!

Q5. Let E be a dense subset of a metric space X, and let f be a uniformly continuous real function defined on E. Prove that f has a continuous extension from E to X. Hint: For each p ∈ X and each positive integer n, let V_n(p) be the set of all q ∈ E with d(p, q) < 1/n. Shoe that the intersection of the closures of the sets F(V₁(p)), f(V₂(p)), . . . , consists of a signal point, say g(p), of R¹. Prove that the function g so defined on X is the desired extension of f.

Could the range space R¹ be replaced by R^k? By any compact metric space? By any complete metric space? By any metric space?

Q6. A real-valued function f defined in (a, b) is said to be convex if

f(λx + (1- λ)y) ≤ λf(x) + (1 - λ)f(y)

where a < x < b, a < y< b, 0 < λ < 1. Prove that every convex function is continuous. Prove that every increasing convex function of a convex function is convex.

If f is convex in (a, b) and if a < s < t < u < b, show that

(f(t)-f(s))/(t-s) ≤ (f(u)-f(s))/(u-s) ≤ (f(u)-f(t))/(u-t).

Q7. Assume that f is a continuous real function defined in a (a, b) such that

f((x+y)/2) ≤ (f(x)+f(y))/2

for all x, y, ∈ (a, b). Prove that f is convex.

Book: Rudin. Principles of Mathematical Analysis. 3th Ed., McGraw-Hill, 1976.

Chapter 4 - Continuity