Question

use the definition of continuity to determine whether f is continuous at a.

Answer 1

I can't read that because i don't have microsoft word. one moment

Answer 2

ok

Answer 3

Dowloading openoffice

Answer 4

ok thank you so much for helping me

Answer 5

apparently the file to download is huge

Answer 6

f(x)= { 2-x if x<1
{ 1 if x=1
{ x^2 if x>1
a=1

Answer 7

for continuity we need to check the points where the function changes, the rest of the function is continuous in its respective parts.

Answer 8

ok how do we do that

Answer 9

Definition of continuity:
Left hand limit = right hand limit = value at the point

Answer 10

We can write this in symbols.

Answer 11

you write it with a little apostrophe thing right

Answer 12

Definition of continuity:
$$ \Large \lim_{x \to a} f(x) = f(a)$$But this implies $$
\Large \lim_{x \to a^{+}} f(x) = \Large \lim_{x \to a^{-}} f(x) =
f(a) $$

Answer 13

So we need to check the left limit and right limits and make sure it equals to the value at that point.

Answer 14

$$ \rm \Large{f(x)= \begin{cases}
2-x ~~~if~ x<1\\
1 ~~~~~~~~ ~\rm ~if~ x=1\\
x^2 \rm ~~~~~~~~if~ x>1
\end{cases}
}
$$

Answer 15

for all the x's i replace them with a 1?

Answer 16

\[ \rm \Large{f(x)= \begin{cases}
2-x ~~~if~ x<1\\
1 ~~~~~~~~ ~\rm ~if~ x=1\\
x^2 \rm ~~~~~~~~if~ x>1
\end{cases}
}
\] \[ \Large \lim_{x \to 1^{+}} f(x) =
\\~\\ \Large \lim_{x \to 1^{-}} f(x) =
\\~\\ \Large f(1) =
\]

Answer 17

yes thats correct. and we need to check that all three of those are equal . thats the continuity condition

Answer 18

so the answer is 1 for all of them.

Answer 19

\[ \rm \Large{f(x)= \begin{cases}
2-x ~~~if~ x<1\\
1 ~~~~~~~~ ~\rm ~if~ x=1\\
x^2 \rm ~~~~~~~~if~ x>1
\end{cases}
}
\] \[ \Large \lim_{x \to 1^{-}} f(x) =\lim_{x \to 1^{-}} 2-x = 2-1 = 1
\\~\\ \Large \lim_{x \to 1^{+}} f(x) = \lim_{x \to 1^{+}} x^2 = 1^2 = 1
\\~\\ \Large f(1) = 1
\]

Answer 20

Thank You So Much,can you help me with one more similar to this one.

Answer 21

ok

Answer 22

determine for what numbers, if any, the given function is discontinuous.

f(x)={x+7 if x is less than or equal to 0
{7 if 0 is <x less than or equal to 3
{x^2-1 if x>3

Answer 23

ok whats the difference between the first set and the second set. (not the questions but what you did)

Answer 24

there are two potential points of discontinuity, where the function abruptly changes.
it changes at x =0 and x = 3.
so we have to test those two parts independently

Answer 25

ok so how do we do that

Answer 26

We can look at the graph that might make it more clear
https://www.desmos.com/calculator/pcjrrf5xdh

Answer 27

edit*
\[ \rm \Large{f(x)= \begin{cases}
x+7 ~~~if~ x\leq0\\
7 ~~~~~~~~ ~\rm ~if~ 0 < x\leq 3\\
x^2-1 \rm ~~if~ x>3
\end{cases}
}
\] \[ \Large \lim_{x \to 0^{-}} f(x) =\lim_{x \to 0^{-}} x+7 = 0+7 = 7
\\~\\ \Large \lim_{x \to 0^{+}} f(x) = \lim_{x \to 0^{+}} 7 = 7
\\~\\ \Large f(0) =0+7 = 7
\] \[ \Large \lim_{x \to 3^{-}} f(x) =\lim_{x \to 3^{-}}7 = 7
\\~\\ \Large \lim_{x \to 3^{+}} f(x) = \lim_{x \to 3^{+}} x^2-1 = 3^2-1 = 8
\\~\\ \Large f(3) = 7
\]

Answer 28

from the graph and from the continuity definition
we see that there is a point of discontinuity at x = 3.
x=0 is fine

Answer 29

so all the question was asking for was two points

Answer 30

Before we do any work the function is potentially discontinuous at the two points x=0 and x = 3 because the function changes suddenly at those points.

But after analyzing the function graphically, or using the limit above, we see that it is actually only discontinuous at one point. x= 3