# Continuity of a Function

Discussion in 'Calculus' started by nycmathguy, May 12, 2022.

1. ### nycmathguy

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Calculus
Section 2.5

I read this twice. It somewhat makes sense to me. During the week, I simply lack the needed time to watch video tube lessons. Now, by continuous I recall Jenn saying that when we are able to draw a graph without lifting the pen or pencil from the piece of paper, the graph is continuous. I understand continuous versus discontinuous as a road. A gap on the road is discontinuous. A smooth road with no holes is continuous for cars to safely drive through. Look at the graph below. At which numbers is the function discontinuous? Explain what you are doing. I want this to make sense to me.

P. S. My \$179 was totally refunded on May 10. Jenn lied. I also don't believe that she has over 15,000 students. This can be a number made up by Jenn to draw idiots like myself. Never again!!!!

nycmathguy, May 12, 2022

2. ### MathLover1

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she has over 15,000 subscribers and
over 12,000 other students
20 Countries Jenn’s students currently span

1.
If f (x ) is continuous at a then lim f(x )=f (a ) as x->a
We see that the graph of f(x) has a hole at x=1.
In fact, f(a) is undefined.
At the very least, for f(x) to be continuous at a, we need the following conditions:

f(a) is defined

2.
In this example, the gap exists because lim(f(x),x-> a does not exist. We must add another condition for continuity at a—namely,

left side lim(f(x)) ,x-> 3 exists, right side lim(f(x)) ,x-> 3 does not exist (there is another hole)

3.
The function f(x) is not continuous at a because lim(f(x)) ≠ f(a) as x-> a

MathLover1, May 13, 2022
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3. ### nycmathguy

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1. What do you mean by f(a) must be defined?

2. What do you mean that the function is discontinuous because the limit of f(x) doesn't equal f(a)?

nycmathguy, May 13, 2022
4. ### MathLover1

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for function to be continuous f(a) must be defined (in your case is not defined because function is not continuous)

f(x) doesn't equal f(a)=> take a look at x=3 , or a=3
f(3)=-2 ( left limit) and f(3)=3 (limit from the right)=> -2 is not equal 3

MathLover1, May 13, 2022
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5. ### nycmathguy

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Where is -2 on the graph? I don't see it. From now on, less questions will be posted. By so doing, our math discussion will be lengthen. I believe more learning takes place by showing math work and thus allowing you to "catch" my error(s) somewhere along the way. If I don't know how to solve a question altogether, I will then make this known to you. So, less questions leading to a deeper, solid, coherent response at both ends. Otherwise, I am wasting precious time.

nycmathguy, May 13, 2022
6. ### MathLover1

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Where is -2 on the graph? -> if you loo closely on the graph where blue point under x=3 is, y-coordinate is actually -1 (not -2), but answer will be same
f(3)=-1 ( left limit) and f(3)=3 (limit from the right)=> -1 is not equal 3

MathLover1, May 13, 2022
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