Let E be closed and bounded in R, and suppose that for each x E there is a function fx, nonnegative, nonconstant, increasing, and C on R, such that fx(x) 0 and fx(y) 0 for y E Prove that there exists a nonnegative, nonconstant, increasing C function f on R such that f(y) 0 for all y E and f(y) 0 fo...

Let x E Since f x x 0 and f x is continuous choose by the Sign Preser ...

Let E be closed and bounded in R, and suppose that for each x E there

Question:

Let E be closed and bounded in R, and suppose that for each x ∈ E there is a function fx, nonnegative, nonconstant, increasing, and C∞ on R, such that fx(x) > 0 and fʹx(y) = 0 for y ∉ E. Prove that there exists a nonnegative, nonconstant, increasing C∞ function f on R such that f(y) > 0 for all y ∈ E and fʹ(y) = 0 for all y ∉ E.