# Continuity at a particular x

1. Jul 27, 2008

### blackisback

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.

2. Jul 27, 2008

### slider142

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. Jul 27, 2008

### tiny-tim

Welcome to PF!

Hi blackisback ! Welcome to PF!

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.

4. Jul 27, 2008

### blackisback

thanks alot guys

5. Jul 28, 2008

### HallsofIvy

Staff Emeritus
I don't think you need to use "epsilon-delta". Just using the one-sided limits should be enough.

6. Aug 2, 2008

### peos69

Is 5 an accumulation point, ???

7. Aug 3, 2008

### slider142

Since the function is continuous there, 5 is an accumulation point of the image of h under the standard topology inherited from R.

8. Aug 3, 2008

### peos69

Since all functions are continuous over an isolated point in there domain your implication is incorrect