S.O.S. Mathematics CyberBoard

If f is continuous on (a,b), the f(a) and f(b) can be defined so that f is integrable on [a,b].

My inclination is to say false and to use sin(1/x) as a counterexample. However, since this is a problem I would need to present on the board using "because" as my explanation probably wouldn't be sufficient. How might I explain this accurately and concisely? Any help appreciated!

MF

_________________
let:
a=b
ab=b^2
ab-a^2=b^2-a^2
a(b-a)=(b+a)(b-a)
a=b+a
a=2a
1=2

I assume you are dealing with Riemann integral rather than the many other definitions of integral?

Recall Lebesgue's characterisation of Riemann integrable functions. {a,b} is null in [a,b], so it doesn't matter what you define f(a), f(b) to be.

_________________

Indeed, you can have a much simpler counter example, use on .

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Everything said above is certainly true...on the other hand, you might be referring to uniform continuity, since we can say something like this:

A function f is uniformly continuous on the interval (a,b) if and only if it can be defined at the endpoints a and b such that the extended function is continuous on [a,b].

This is sometimes called the "Continuous Extension Theorem". The essential part of the argument to verify it is that if we assume f to be uniformly continuous and the sequence {x_k} is Cauchy, then the image sequence {f(x_k)} is also Cauchy.

Therefore, you may choose to rephrase the original question as follows:

_________________
"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
G.H. Hardy (1877-1947)

Indeed, and actually sin(1/x) is not a counterexample since it is bounded near 0.

_________________