Let f AX be a function such that A CX. Show that if A is compact, then the range of f defined by f(A) = {f(x): € A} is bounded.

We can express the function in Exercise 3 algebraically as f(x)=g(x)= (fg)(x)=(gf)(x)=

Determine whether the statement " If f is a function, then f(ax, ay) = a2f(x, y)" ,is true or false. If it is false, explain why or give an example that shows it is false.

if f functions is bounded on [a,b] does f has an absolute maximum and absolute minimum or not??

Let f:[0,1]→R, be the function defined by, f(x)=|x-1/2|. Show that f is continuous and conclude that it is also absolutely continuous. Please be as clear as possible and use definitions if necesary

What does it mean to say f : (a, b) → R is a bounded function? If a < b in R and f : (a,b) → R is uniformly continuous, prove it is a bounded function. Give examples of two functions f,g : R → R which are both uniformly continuous and where f is not bounded but g is bounded.

Determine whether the statement " If f is continuous for all x and y and has two relative minima, then f must have at least one relative maximum ", is true or false. If it is false, explain why or give an example that shows it is false.

Suppose that a 0. Prove f(T) is bounded. that the function } : [a, b] → R defined by () (x) = =

c) Evaluate f x*(In x)²dx