Topology question:
Show that a function f : ℝ → ℝ is continuous in the ε − δ
definition of continuity if and only if, for every x ∈ ℝ and every
open set U containing f(x), there exists a neighborhood V of x such
that f(V) ⊂ U.
For the following exercises, determine whether or not the given function f is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.f(x) = sec(x) − 3.
Show all your work. Let f(x) = x 5 e x 3 (i) Use the Taylor
series for e x around 0 to find the Taylor series for f(x) (ii) Use
(i) to find f (20)(0), f (21)(0), f (22)(0), f (23)(0)
1) Show the absolute value function f(x) = |x| is continuous at
every point.
2) Suppose A and B are sets then define the cartesian product A
* B
Please answer both the questions.
f(x)=0 if x≤0, f(x)=x^a if x>0
For what a is f continuous at x = 0
For what a is f differentiable at x = 0
For what a is f twice differentiable at x = 0
Define a function ?∶ ℝ→ℝ by
?(?)={?+1,[?] ?? ??? ?−1,[?]?? ????
where [x] is the integer part function. Is ? injective?
(b) Verify if the following function is
bijective. If it is bijective, determine its inverse.
?∶ ℝ\{5/4}→ℝ\{9/4} , ?(?)=(9∙?)/(4∙?−5)