Suppose we have a function f from R to R that is continuous on (a,b]. Define g by g(x) = f(x) if x <> a and g(a) = lim f(x) as x approaches a. Is it true that g is continuous on [a,b]?(adsbygoogle = window.adsbygoogle || []).push({});

I would think it is, but I'm having a hard time proving it. I'm trying to use sequences to do this: Suppose {s_n} is a sequence in [a,b] that converges to s.

If s <> a, then then {s_n} contains only finitely many a's, whence there exists an M such that s_n <> a for all n > M. Thus, the sequence {s_m} where m > M contains no a's, so {g(s_m) = f(s_m)} converges to f(s) = g(s) by the continuity of f, i.e. for any e > 0, there is an N such that |g(s_m) - g(s)| < e for all n > N. Thus, |g(s_n) - g(s)| < e for all n > N > M, so g is continuous at s <> a.

Now suppose s = a. If {s_n} contains finitely many a's, then I can use the same strategy as mentioned above to conclude that {g(s_n)} converges to a. If {s_n} contains infinitely many a's, then surely there's a subsequence {s_m} of {s_n} such that {g(s_m)} converges g(a). How can I extend this so that {g(s_n)} converges to g(a)?

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# Simple Extension of a Function

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