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# Continuity of a function in [a,b] Watch

1. I'm stuck with part b) but I'm not sure if I've done a) correctly neither.

For a) I took the left and right hand limits 'a-' and 'a+' and deduced that both ----> 0 and so the limit of 'a' must also go to 0, suggesting the function is continuous at x=a and I did this for 'b-' and 'b+' too also meaning that the function must be continuous at x=b too! Is this the correct approach?
2. (Original post by As_Dust_Dances_)

I'm stuck with part b) but I'm not sure if I've done a) correctly neither.

For a) I took the left and right hand limits 'a-' and 'a+' and deduced that both ----> 0 and so the limit of 'a' must also go to 0, suggesting the function is continuous at x=a and I did this for 'b-' and 'b+' too also meaning that the function must be continuous at x=b too! Is this the correct approach?
The working for the limits "a-" and "b+" are outside the domain of f, and don't come into it.

Although I'm not an analysis expert, I'd say (x-a) is continuous on [a,b], as is (b-x), therefore their product is also. Square rooting is a continuous function (perhaps iffy in an analysis course), and (x-a)(x-b) is in the domain of that function, hence the whole thing is continuous on [a,b].

For the second one f is not defined at x=b. Aside from that proceed as per first one.

I'm sure someone more on the ball will correct me if I'm in error here.
3. (Original post by ghostwalker)
The working for the limits "a-" and "b+" are outside the domain of f, and don't come into it.

Although I'm not an analysis expert, I'd say (x-a) is continuous on [a,b], as is (b-x), therefore their product is also. Square rooting is a continuous function (perhaps iffy in an analysis course), and (x-a)(x-b) is in the domain of that function, hence the whole thing is continuous on [a,b].

For the second one f is not defined at x=b. Aside from that proceed as per first one.

I'm sure someone more on the ball will correct me if I'm in error here.
I was actually thinking along the lines of the "product" such as splitting it up into f(x) and g(x) then using the lim x --> a, but I'm not certain. I'm planning on seeing my lecturer tomorrow about it, I think I understand the basics of continuity, it's just questions like these that throw me off slightly and whether or not my lecturer expects me to do it some specific way I'm not sure. Thanks for your help anyway! (Won't let me give you rep)
4. It's a little hard without context (i.e. some idea of how long you're supposed to spend on the question).

For a bit of "light discussion", what ghostwalker says is fine. But if the exam paper implies it's, say, 20 minutes work, then I would think a proper epsilon-delta discussion would be more appropriate.
5. The other thought that struck me is uniform continuity, or not, as the case may be.
6. (Original post by ghostwalker)
The other thought that struck me is uniform continuity, or not, as the case may be.
I think that's unlikely; a continuous function on a closed interval is always uniformly continuous.
7. (Original post by DFranklin)
a continuous function on a closed interval is always uniformly continuous.
Granted, but wadr the second function is only defined on [a,b)

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Updated: April 18, 2013
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