함수의 연속

[T] After a certain distance D has passed, the gravitational effect of Earth becomes quite negligible, so we can approximate the force function by $F(d)=\left\{\begin{array}{ll}-\frac{mk}{{d}^{2}} & \text{if}\,d<D \\ 10,000 & \text{if}\,d\ge D\end{array}\right..$ Using the value of k found in the previous exercise, find the necessary condition D such that the force function remains continuous.

As the rocket travels away from Earth’s surface, there is a distance D where the rocket sheds some of its mass, since it no longer needs the excess fuel storage. We can write this function as $F(d)=\left\{\begin{array}{l}-\frac{{m}_{1}k}{{d}^{2}}\,\text{if}\,d<D \\ -\frac{{m}_{2}k}{{d}^{2}}\,\text{if}\,d\ge D\end{array}\right..$ Is there a D value such that this function is continuous, assuming ${m}_{1}\ne {m}_{2}?$

Prove the following functions are continuous everywhere

$f(\theta )=\sin \theta$

Prove the following functions are continuous everywhere

$g(x)=|x|$

Prove the following functions are continuous everywhere

Where is $f(x)=\left\{\begin{array}{l}0\,\text{if}\,x\,\text{is irrational} \\ 1\,\text{if}\,x\,\text{is rational}\end{array}\right.$ continuous?

In your own words, describe what it means for a function to be continuous.

In your own words, describe what the Intermediate Value Theorem states.

What is a “root” of a function?

Given functions $f$ and $g$ on an interval $I$, how can the Bisection Method be used to find a value $c$ where $f(c) = g(c)$?

T/F: If $f$ is defined on an open interval containing $c$, and $\displaystyle \lim_{x\to c}f(x)$ exists, then $f$ is continuous at $c$.

T/F: If $f$ is continuous at $c$, then $\displaystyle \lim_{x\to c}f(x)$ exists.

T/F: If $f$ is continuous at $c$, then $\displaystyle \lim_{x\to c^+}f(x) = f(c)$.

T/F: If $f$ is continuous on $[a,b]$, then $\displaystyle \lim_{x\to a^-}f(x) = f(a)$.

T/F: If $f$ is continuous on $[0,1)$ and $[1,2)$, then $f$ is continuous on $[0,2)$.

T/F: The sum of continuous functions is also continuous.

In Exercises 11–18, a graph of a function $f$ is given along with a value $a$. Determine if $f$ is continuous at $a$; if it is not, state why it is not.

$a = 1$

In Exercises 11–18, a graph of a function $f$ is given along with a value $a$. Determine if $f$ is continuous at $a$; if it is not, state why it is not.

$a = 1$

In Exercises 11–18, a graph of a function $f$ is given along with a value $a$. Determine if $f$ is continuous at $a$; if it is not, state why it is not.

$a = 1$

In Exercises 11–18, a graph of a function $f$ is given along with a value $a$. Determine if $f$ is continuous at $a$; if it is not, state why it is not.

$a = 0$

In Exercises 11–18, a graph of a function $f$ is given along with a value $a$. Determine if $f$ is continuous at $a$; if it is not, state why it is not.

$a = 1$