Continuity at a particular x

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In summary, the conversation involves a request for help with a problem involving proving the continuity of a function h(x) at x=5. The participants discuss using the definition of continuity and the use of one-sided limits to prove this. They also consider whether 5 is an accumulation point of the function and clarify that all functions are continuous over an isolated point in their domain.
  • #1
blackisback
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Can anyone help me with this problem?

Say f(x) & g(x) are cont. at x=5.
Also that f(5)=g(5)=8.

If h(x)=f(x) when x<=5
and
h(x)=g(x) when x>=5:

prove h(x) is cont at x=5.
 
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  • #2
Hmm, this appears to be a straightforward application of the definition of continuity at a point. You already know the h(5) exists, now you just need to show that the limit as x approaches 5 of h(x) is 8. This can be done as 2 particular cases, as h(x) either =f(x) or g(x), depending on which side you approach the point from.
 
  • #3
Welcome to PF!

blackisback said:
Can anyone help me with this problem?

Say f(x) & g(x) are cont. at x=5.
Also that f(5)=g(5)=8.

If h(x)=f(x) when x<=5
and
h(x)=g(x) when x>=5:

prove h(x) is cont at x=5.

Hi blackisback ! Welcome to PF! :smile:

Just write out the definitions of:

f(x) is continuous at x = 5
g(x) is continuous at x = 5
h(x) is continuous at x = 5

using the δ and ε method.

Then just chug away. :smile:
 
  • #4
thanks a lot guys
 
  • #5
I don't think you need to use "epsilon-delta". Just using the one-sided limits should be enough.
 
  • #6
Is 5 an accumulation point, ?
 
  • #7
peos69 said:
Is 5 an accumulation point, ?

Since the function is continuous there, 5 is an accumulation point of the image of h under the standard topology inherited from R.
 
  • #8
Since all functions are continuous over an isolated point in there domain your implication is incorrect
 

1. What does it mean for a function to be continuous at a particular x?

Continuity at a particular x means that the function is defined at that value of x and the limit of the function as x approaches that value is equal to the value of the function at that point.

2. How can I determine if a function is continuous at a particular x?

To determine if a function is continuous at a particular x, you can use the continuity test which states that if the limit of the function as x approaches the value in question exists and is equal to the value of the function at that point, then the function is continuous at that value of x.

3. What is a removable discontinuity at a particular x?

A removable discontinuity at a particular x occurs when the limit of the function as x approaches the value in question exists, but is not equal to the value of the function at that point. This means that there is a hole or gap in the graph of the function at that point.

4. Can a function be continuous at a particular x but not differentiable at that point?

Yes, a function can be continuous at a particular x but not differentiable at that point. This means that although the function is defined and has a limit at that point, the function is not smooth and does not have a well-defined tangent line at that point.

5. How can I determine if a function is continuous on a closed interval?

To determine if a function is continuous on a closed interval, you can use the Intermediate Value Theorem. This theorem states that if a function is continuous on a closed interval and the function takes on two different values at the endpoints of the interval, then the function must take on every value in between those two values on the interval.

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