Show that f(x) = 1/x A) is continuous at x=2 B) is uniformily continuous for [1, infinity) but not for (0,1)

Show that f = {(x, y) ∈ R × R; x = y^3} is a function from R to R

Show that f is not continuous at (0, 0)

Examine if f(x) = x3sin(1/x) is uniform continuous on the interval (0,2]

f" > 0 on (-∞, -1) U (-1,0).

Assume that f: R→ R and g: R → R are infinitely differentiable functions. Prove the following generalization of the product rule: dn dxn n (f(z)g(2)) - (1) (M) (2) g-) (x)}) = k=0

, by definition not direct

We are given the following functionsf : Z → Z × Z, f(x) = (x^2, x^3). (a) Find images f({1, 2, 3}) and f(N).(b) Find those preimages f^−1({1, 2, 3, 4}) and f^−1({(1, −1),(4, 8)}) that are welldefined.

Estimate f(x) dx given that f is a continuous function containing (-2, 8), (–1,0), (4, 125) , and points of the form (x, e") for all positive integers x using an appropriate Newton-Cotes quadrature rule utilizing all the given data in this problem.

Show that A Function with a Single Point of Discontinuity 2xy x² +1,²² 0, is continuous at every point except the origin f(x, y) = (x, y) = (0,0) (x, y) = (0, 0)