Is the exponential function continuous in ${\mathbb{R}}^{n}$? How can one show this? Let's say for a function like $f:{\mathbb{R}}^{2}\u27f6\mathbb{R},f(x,y)={x}^{y}$

Lisantiom
2022-10-05
Answered

Is the exponential function continuous in ${\mathbb{R}}^{n}$? How can one show this? Let's say for a function like $f:{\mathbb{R}}^{2}\u27f6\mathbb{R},f(x,y)={x}^{y}$

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Johnny Parrish

Answered 2022-10-06
Author has **12** answers

The domain is not the entire ${\mathbb{R}}^{2}$, as we have trouble with negative values of x. If x>0, we have that ${x}^{y}={e}^{y\mathrm{log}x}$, which is continuous because the one-variable exponential and $(x,y)\mapsto y\mathrm{log}x$ are continuous.

asked 2022-10-18

What is example of functions that grow faster than the exponential functions and/or factorial functions?

asked 2022-10-13

I'm trying to derive the following two inequalities for the exponential function, where $x,y,z\in \mathbb{R}$, t>0 and a>1

$$|y-z|{e}^{-\frac{(y-z{)}^{2}}{2at}}\le \phantom{\rule{thickmathspace}{0ex}}{C}_{a}{t}^{\frac{1}{2}}{e}^{-\frac{(y-z{)}^{2}}{4at}},\phantom{\rule{0ex}{0ex}}{t}^{-\frac{1}{a}}|y-z{|}^{1+\frac{2}{a}}{e}^{-\frac{(y-z{)}^{2}}{2at}}\le \phantom{\rule{thickmathspace}{0ex}}{C}_{a}{t}^{\frac{1}{2}}{e}^{-\frac{(y-z{)}^{2}}{4at}},$$

with some constant ${C}_{a}>0$

I tried to derive it by using derivatives but it didn't work for the purpose.

$$|y-z|{e}^{-\frac{(y-z{)}^{2}}{2at}}\le \phantom{\rule{thickmathspace}{0ex}}{C}_{a}{t}^{\frac{1}{2}}{e}^{-\frac{(y-z{)}^{2}}{4at}},\phantom{\rule{0ex}{0ex}}{t}^{-\frac{1}{a}}|y-z{|}^{1+\frac{2}{a}}{e}^{-\frac{(y-z{)}^{2}}{2at}}\le \phantom{\rule{thickmathspace}{0ex}}{C}_{a}{t}^{\frac{1}{2}}{e}^{-\frac{(y-z{)}^{2}}{4at}},$$

with some constant ${C}_{a}>0$

I tried to derive it by using derivatives but it didn't work for the purpose.

asked 2022-10-15

Why is the exponential function injective but not surjective?

asked 2022-11-05

How to change the form of an exponential function. Can someone explain the process in which the function below

$$T(t)={e}^{-0.0407409t+3.89467}+26$$

is changed into this form.

$$T(t)=49.1398{e}^{-0.0407409t}+26$$

$$T(t)={e}^{-0.0407409t+3.89467}+26$$

is changed into this form.

$$T(t)=49.1398{e}^{-0.0407409t}+26$$

asked 2022-09-03

$$\underset{p\to 0}{lim}\frac{1}{2p}((1+p){e}^{-\frac{y}{1+p}}-(1-p){e}^{-\frac{y}{1-p}})={e}^{-y}+y{e}^{-y}$$

I have already tried L'Hopital's Rule, but it gave me something that I couldn't simplify. The problem seems to be the $\frac{1}{2p}$ term never seems to go away. I know the exponential function can be represented as: ${e}^{x}=\underset{n\to \mathrm{\infty}}{lim}(1+\frac{x}{n}{)}^{n}$, but it doesn't seem immediately obvious how that would apply in this situation.

I have already tried L'Hopital's Rule, but it gave me something that I couldn't simplify. The problem seems to be the $\frac{1}{2p}$ term never seems to go away. I know the exponential function can be represented as: ${e}^{x}=\underset{n\to \mathrm{\infty}}{lim}(1+\frac{x}{n}{)}^{n}$, but it doesn't seem immediately obvious how that would apply in this situation.

asked 2022-07-15

How do I find vertical asymptotes using an exponential function? ($f(x)=a{b}^{x-h}+k$)I know that horizontal asymptotes are y = k, but I don't know how to find the vertical ones

asked 2022-11-15

Is there a way to transform the function

exp(A+B+C),

where $\mathrm{exp}(\cdot )$ is the exponential function, into a sum

f(A)+f(B)+f(C)?

exp(A+B+C),

where $\mathrm{exp}(\cdot )$ is the exponential function, into a sum

f(A)+f(B)+f(C)?