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title: "Differentiable! \nYou're continuously killing me!"

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Math 181 Miniproject 3: Texting Lesson.md
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Differentiable! 
You're continuously killing me!
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<div><img class="left"/><div class="alert gray">

Hey, you look a little confused... do you need help on a problem?

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<div><div class="alert blue">

YES! Please! I can't seem to get the concepts of a function being differentiable and a function being continuous straight! The more I read, the more lost I get! 
:(  
 
</div><img class="right"/></div> 


<div><img class="left"/><div class="alert gray">

No worries. Let me first explain what a differentiable function entails and what a continuous function is. A function is **DIFFERENTIABLE** so long as there is an existing derivative for each point comprised within that function's domain. In order for a function to be **CONTINUOUS**, it must adhere to the three following conditions: 
*1. f is defined at a.*
*2. f has a limit as x approaches a.*
3.$\lim\limits_{x\to a}$ f(x) = f(a)

This ultimately results in a function that, within its domain, when sketched, it is done in one continuous motion. The correlation between these two concepts is when a function is continuously differentiable, occurring when f is differentiable and its derivative is continuous.  
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<div><div class="alert blue">

Can you give me an example of each one on a graph?

</div><img class="right"/></div> 

<div><img class="left"/><div class="alert gray">

Sure!
So here we have an example of a graph of a function that is continuous:
![](https://i.imgur.com/FaL7Fhz.png)


And here is a graph of a function that is differentiable:
![](https://i.imgur.com/xbqVco4.png)






</div></div> 

<div><div class="alert blue">

What would be an example of a continuously differentiable function when graphed?


</div><img class="right"/></div>

<div><img class="left"/><div class="alert gray">

A continuously differentiable function would look like this:
![](https://i.imgur.com/5GkVqJu.png)



</div></div> 

<div><div class="alert blue">

Oh I see! Thanks for for the help! 

</div><img class="right"/></div>


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