Many mathematicians prefer to define the continuity of a function via a so-called epsilon-delta definition of a limit. In this formalism, a limit of function as approaches a point , (1) is defined when, given any , a can be found such that for every in some domain and within the ...
Use the epsilon-delta definition of continuity to show that f(x) = x^2 + 7x is continuous on R. Given f(x) = \sqrt{x-4} show is not continuous at 4 and determine continuity on [4,13] . Let f,g: R\rightarrow ...
Specifically, we show that given a continuous function f : M1 → M2 from a metric space (M1, d1) to a metric space (M2, d 2), there is a function delta : M1 x R+→R+ such that delta is continuous and, if ∀x ∈ M1 ∀epsilon > 0 ∀y ∈ M1 [d1(x, y) < delta(...
that is, the only root off(y,\cdot ), belonging to{\overline{B}}_{\varepsilon _1} (w_j). Since the boundary\partial B_{\varepsilon _1}(w_j)ofB_{\varepsilon _1}(w_j)is compact,\delta _j:=\min \{\vert f(y,z)\vert :z\in \partial B...
I have been trying just the basic epsilon-delta definition way (similar to what I did in part (a) with this direction) and not really getting anywhere (which I guess could be expected since we don't have a uniform continuity definition with sequences [at least in our book])....
We say f is \varepsilon-close to L near x_0 if and only if there exists a \delta>0 such that \left| f(x)-L \right| \le \varepsilon when restricted to the set \{x\in X : |x-x_0|<\delta\}. [Definition 9.3.6] (Convergence of function at a point). Let X be a subset...
Use the delta-epsilon proof to show that f(x) = (x - 1) / (x + 1) is uniformly continuous on [0,\infty). If f is continuous on (0, 1), prove that integral_{0}^{1} f(x) dx = integral_{0}^{1} f(1-x) dx. Let f be a function continuous on a closed interva...
\(\liminf_{t\to \infty }(E(t)+I(t))>\varepsilon \) for any solution with \(E(0)+I(0)>0\). If lim inf is substituted by lim sup in the definition, the system is said to be uniformly weakly persistent. In the next theorem we prove the uniform persistence of system (4...
Social interactions evolve continuously. Sometimes we cooperate, sometimes we compete, while at other times we strategically position ourselves somewhere in between to account for the ever-changing social contexts around us. Research on social interactio
Let us assume that the amplitude of our signal is small, say\(|q(t)| \sim \varepsilon\), with\(\varepsilon \ll 1\). Then, we can derive the following expansions for the NF scattering coefficients14: $$\begin{aligned} a(\xi ) = 1 - \intop _{-\infty }^{\infty } dt_1 \...