Proof: Show that f(x) is continuous at a

[shwanky]

I'm a little nervous about a test I have on Thursday and I was wondering if this is adequate for a proof of the following equation.

Homework Statement

Show that f(x) is continuous at a=4

Homework Equations

$$f(x) = x^2 + \sqrt{7-x}$$

The Attempt at a Solution

For f(x) to be continuous at a = 4
a) f(a) must be defined

b) $$\lim_{x \to a} f(x)$$ must exist

c) $$\lim_{x \to a} f(x) = f(a)$$

Proof

1. $$f(a) = f(4) = 4^2 + \sqrt{7-4} = 16 + \sqrt{3}$$
Therefore, f(a) is defined at a=4.

2. $$\lim_{x \to a} f(x) = \lim_{x \to 4} x^2 + \sqrt{7-x}$$

$$\lim_{x \to 4^+} x^2 + \sqrt{7-x} = 16 + \sqrt{3}$$

$$\lim_{x \to 4^-} x^2 + \sqrt{7-x} = 16 + \sqrt{3}$$

Since $$\lim_{x \to 4^+} x^2 + \sqrt{7-x} = 16 + \sqrt{3} = \lim_{x \to 4^-} x^2 + \sqrt{7-x} = \lim_{x \to 4} f(x)$$ exists.

$$\lim_{x \to 4} x^2 + \sqrt{7-x} = 16 + \sqrt{3}$$

3. By steps 1 and 2 we know that f(4) = $$16 + \sqrt{3}$$ and $$\lim_{x \to 4} x^2 + \sqrt{7-x} = 16 + \sqrt{3}$$.

Since $$\lim_{x \to 4} f(x) = 16 + \sqrt{3} = f(4)$$, $$f(x) = x^2 + \sqrt{7-x}$$ is continuous at a = 4.

 

[eaboujaoudeh]

adequate? more than enough,in my times if a curve went to the same number from the -ve side and the +ve side, then it would be continous:)

 

[matt grime]

That is not adequate. State your definition of continuity and show it is true. I saw nothing there that constitutes a proof. Nothing there even acknowledges the definition of continuous so it cannot possibly be a proof.

 

[shwanky]

Do I have to write the definition each time I prove the continuity of an equation, or rather should I?

 

[shwanky]

According to my book, a function f is said to be continuous at $$\lim_{x \to a} f(x) = f(a)$$. Following this definition it says that that a function is continuous at a point if a, b and c are true :-/.

 

[ZioX]

According to my book, a function f is said to be continuous at $$\lim_{x \to a} f(x) = f(a)$$. Following this definition it says that that a function is continuous at a point if a, b and c are true :-/.

http://video.google.ca/videoplay?docid=8266735963828183844&q=how+to+write

Your last line reminded me of this. :)

 

[shwanky]

sad part is, at one point I was considering being an english major... MUYAY!

 

[matt grime]

According to my book, a function f is said to be continuous at $$\lim_{x \to a} f(x) = f(a)$$. Following this definition it says that that a function is continuous at a point if a, b and c are true :-/.

What are a,b and c? What does it mean for f(x) to tend to f(a)? Are we looking for a proper epsilon/delta proof, or does your course not require proofs?

 

[HallsofIvy]

2. $$\lim_{x \to a} f(x) = \lim_{x \to 4} x^2 + \sqrt{7-x}$$

$$\lim_{x \to 4^+} x^2 + \sqrt{7-x} = 16 + \sqrt{3}$$

$$\lim_{x \to 4^-} x^2 + \sqrt{7-x} = 16 + \sqrt{3}$$

But how do you get those last two limits? Not by just setting x= 4, surely, since that requires knowing that the function is continuous at 4- exactly what you are trying to prove!

 

[HallsofIvy]

What are a,b and c?

Actually, he does have a definition for "a". Of course, he should explain what it means for a number to be "true"!

 

[shwanky]

But how do you get those last two limits? Not by just setting x= 4, surely, since that requires knowing that the function is continuous at 4- exactly what you are trying to prove!

There's a theorem in my book which states a $$\lim_{x \to a} = L$$ exists if and only if $$\lim_{x \to a^+} f(x) = L = \lim_{x \to a^-} f(x)$$ unfortunately, my book doesn't substatiate this with a proof. It just states the theorem and my professor didn't bother proving this.

Are we looking for a proper epsilon/delta proof, or does your course not require proofs?

I'm not exactly sure what your asking so my answer would be, at this point no?

 

[Edgardo]

Hello shwanky,

do you know any other theorems about continuous functions?
Maybe you can use some from here:
http://www.libraryofmath.com/properties-of-a-continuous-functions.html

 

[dontdisturbmycircles]

There's a theorem in my book which states a $$\lim_{x \to a} = L$$ exists if and only if $$\lim_{x \to a^+} f(x) = L = \lim_{x \to a^-} f(x)$$ unfortunately, my book doesn't substatiate this with a proof. It just states the theorem and my professor didn't bother proving this.



I'm not exactly sure what your asking so my answer would be, at this point no?

Just take it to be a defintion of a limit, the limit exists if and only if the two one-sided limits exist and are equal.

Let f be a function defined on an open interval containing c (except posibly at c) and let L be a real number. The statement

$$\lim_{x \to c} f(x) = L$$

means that for each $$\epsilon > 0$$ there exists a $$\delta > 0$$ such that if

$$0 < |x-c| < \delta$$, then $$|f(x) - L| < \epsilon$$

which basically says if for any small positive number $$\epsilon$$ with which you say "Can we make f(x) lie in the interval (L-$$\varepsilon$$ , L+$$\varepsilon$$)" by finding a number $$\delta$$ such that we make x lie in the interval (c-$$\delta$$, c+$$\delta$$) for all positive $$\varepsilon$$If the answer is yes then the limit is L as x approaches c and if f(c)=L then the function is continuous at c.

But that's the formal definition of the limit, and might not be used in high schools for example. But I think that is how you ussually prove continuity, with an epsilon-delta proof... But if you haven't seen this stuff before, maybe phone a friend ( smart one preferably :P ) and ask if you are supposed to use the regular limit laws to 'prove' the limits exist.

 

[dontdisturbmycircles]

If you don't mind me asking, what textbook are you using? Epsilon and delta represent small positive numbers in this scenario. (but they are just variables, right?)

(this part was in reference to a question posed which has since been deleted..)

Try looking at some of these if you need help with epsilon-delta proofs:

http://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/preciselimdirectory/PreciseLimit.html
http://lobe.ibme.utoronto.ca/mat196f/epsilon-delta-proofs.htm

And if you have questions about it feel free to ask.

(edit: and make sure its neccesary for your course. Although the precise definition of a limit gave me a better idea of what a limit really is, and it probably won't hurt to learn it. If you don't need it with your course, just show through the regular limit laws (the ones you were using in your first post) that the left and right hand limits exist and are equal to L, and that f(c)=L) In your proof maybe suggest why you are finding the limits (to prove that both one-sided limits exist and are equal) and such...

 

[las3rjock]

Is this a freshman-level course or a upper-level undergraduate course? If it is a freshman-level course, then your proof is probably adequate, but if it is an upper-level undergraduate course, then you probably should use epsilon-delta to prove the limit.

 

[shwanky]

freshman... for the love of God freshman ^.^... hehe... I hope at some point this thing starts making some sense...

 

[dontdisturbmycircles]

Yea, keep in mind that I was just introducing you to the idea of a delta-epsilon proof because you seemed interested in what it was.

edit: then you were on the right path (if you don't need the delta-epsilon 'proof') :) Just maybe mention in your proof what you are doing and why before you do it.

 

[shwanky]

I'm currently using Stewart "Calculus" 5th ed, Brooks/Cole. I dont' know if we get to epsilon delta proofs further along in the book, but for now they aren't discussed in any of the chapters that have been assigned.

 

[dontdisturbmycircles]

I think its in the appendix. Its optional.

Actually I have a copy and mine introduces it right away, but I know there are other copies where its in the appendix. :P

If you don't need it, don't do it with delta-epsilon proofs. I don't know what you need to do, only you do, or at least should. :P If your teacher hasn't even mentioned delta-epsilon proofs, I'd say your safe to just apply your limit laws. Anyways, I am out, good luck. Cya.

 

[shwanky]

I can't believe I have 3 years of this... -.-, I think I'm going to die. lol

 

[dontdisturbmycircles]

Its not that bad, really. At first I was in the same mindset as you... All these new symbols and concepts. But once you get going it starts making sense. Its a hard course that's for sure, just MAKE SURE not to fall behind... or your f-----'d lol :) If you keep up, you'll be fine. I can't speak of the later calc courses (haven't done them) but for calc I and II I can speak that once you get the ideas of limits and derivatives/integrals down its all just applications and not that hard. I am trying to self - learn the last bit of calc II now and then moving on to calc III :P

 

[shwanky]

It's not so much the symbols or the concepts that are hard, it's the explanation... I'm trying to understand my professors Korean-English while attempting comprehend the course work. He's a nice guy and I don't mind him as a teacher. But sometimes his explanation is more confusing than the thing he's trying to explain...

 

[HallsofIvy]

There's a theorem in my book which states a $$\lim_{x \to a} = L$$ exists if and only if $$\lim_{x \to a^+} f(x) = L = \lim_{x \to a^-} f(x)$$ unfortunately, my book doesn't substatiate this with a proof. It just states the theorem and my professor didn't bother proving this.

That was not my point. If you are using that to prove that the limit exists, you can't just assert that the two one-sided limits exist- you have to show that!

Are we looking for a proper epsilon/delta proof, or does your course not require proofs?

I'm not exactly sure what your asking so my answer would be, at this point no?

You are asked to SHOW something is true. It might not be an "epsilon-delta" proof but you are still being asked for a proof!

 

[shwanky]

How exactly would I show that they are true then?

 

[ZioX]

You should probably justify that the sum of two continuous functions is continuous. And then make some comment on how the composition of continuous functions are continuous (where composition works). And then it's trivial if you realize that exponentiation is continuous.

 

[matt grime]

How exactly would I show that they are true then?

That is up to you to decide given the requirements of your course. Look at examples done by the teacher - how do they prove things? Do they make appeals to things like 'the sum of continuous functions is continuous'?

 

[shwanky]

My professor just shows

a) f(a) must be defined

b) $$\lim_{x \to a} f(x)$$ must exist

c) $$\lim_{x \to a} f(x) = f(a)$$

I looked in my book and the section which uses the epsilon delta proof was skipped... He said it's to complicated to for us right now and will come back to it at the end of the semester... But that really just makes this more confusing :-/...