document.write( "Question 15389: Let f: R to R be a function. Show that if f is continuous on
\n" ); document.write( "[0,infinitive) and uniformly continuous on [a,infinitive) for some
\n" ); document.write( "positive real number a, then f is uniformly continuous on [0,infinitive)
\n" ); document.write( "(f is uniformly continuous on [0,b] for any b>a).
\n" ); document.write( "Thankyou.
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Algebra.Com's Answer #7625 by khwang(438)\"\" \"About 
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Let f: R to R be a function. Show that if f is continuous on
\n" ); document.write( "[0,infinitive) and uniformly continuous on [a,infinitive) for some
\n" ); document.write( "positive real number a, then f is uniformly continuous on [0,infinitive)
\n" ); document.write( "(f is uniformly continuous on [0,b] for any b>a).
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\n" ); document.write( " First of all, I wonder why you ask this of calculus or real analysis here.\r
\n" ); document.write( "\n" ); document.write( " It seems you should use the theorem that if a function is
\n" ); document.write( " continuous on a compact set C(closed and bounded set in R),then
\n" ); document.write( " f is continuous on C.\r
\n" ); document.write( "\n" ); document.write( " Proof: Now f is continuous on[0,+oo), so as a subset,
\n" ); document.write( " f is continuous on [0,a] and thus unif. conti. on [0,a].
\n" ); document.write( " Also, we know that f is unif. conti. on [a,+oo).
\n" ); document.write( " But, [0,+oo) = [0,a] U [a,+oo) , we conclude that f is unif. conti. on [0,+oo).\r
\n" ); document.write( "\n" ); document.write( " More precisely, since f is unif. conti. on [a,+oo)
\n" ); document.write( "for any \"epsilon\" > 0, there exists \"delta%5B1%5D\" > 0 such that
\n" ); document.write( " |x-y|< \"delta%5B1%5D\" implies {f(x) -f(y)| < \"epsilon%2F2\"
\n" ); document.write( " for all x, y in [a,+oo)
\n" ); document.write( " And, f is unif. conti. on [0,a]. for the same \"epsilon\",
\n" ); document.write( " there exists \"delta%5B2%5D\" > 0 such that
\n" ); document.write( " |x-y|< \"delta%5B2%5D\" implies |f(x) -f(y)| < \"epsilon%2F2\"
\n" ); document.write( " for all x, y in [0,a].\r
\n" ); document.write( "\n" ); document.write( " Choose \"delta\" =min (\"delta+%5B1%5D\", \"delta+%5B2%5D\") the we have
\n" ); document.write( " |x-y| < \"delta\" implies |f(x) -f(y)| < \"epsilon%2F2\"
\n" ); document.write( " for all x,y in [0,a] (or [a,+oo))
\n" ); document.write( " if x is in [0,a] and y is in [a,+oo) we have (Note:a btwn x & y)
\n" ); document.write( " |x-y| < \"delta\" implies |x-a| < \"delta\" and |y-a| < \"delta\"
\n" ); document.write( " and hence |f(x) - f(y)| <= |f(x) - f(a)| + |f(a) - f(y)|
\n" ); document.write( " < \"epsilon%2F2\" + \"epsilon%2F2\" = \"epsilon\"
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\n" ); document.write( " This shows f is unif. conti.on [0,+oo)\r
\n" ); document.write( "\n" ); document.write( " Try to read carefully and draw diagram to understand the details.\r
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\n" ); document.write( "\n" ); document.write( " Kenny
\n" ); document.write( " [Note: you don't have to worry about the interval, [0, b] for any b.
\n" ); document.write( " Since [0,b] is compact subset of R. However, f is unif. conti. on [0,b]
\n" ); document.write( " for any b > 0 does not imply f is unif. conti. on [0,+oo) as
\n" ); document.write( " the example f(x) = \"x%5E2\" on [0,+oo) shows.
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