document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184440>Question 1184440</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Create a function piecewise and verify that the function is uniform throughout the domain and has continuity.\r<BR>\n" );
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document.write( "f(x)={g(x), if 0≤x≤a<BR>\n" );
document.write( "-----{w(x), if a < x < b<BR>\n" );
document.write( "-----{h(x), if b≤x≤c<BR>\n" );
document.write( "where w (x) must be equal to one of the following functions:<BR>\n" );
document.write( "e^(x^2)<BR>\n" );
document.write( "sen(x^2)<BR>\n" );
document.write( "cos(x^2)<BR>\n" );
document.write( "ln(x^2)<BR>\n" );
document.write( "sen(e^x) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #815043 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=815043\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> One such function is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+system%28matrix%283%2C2%2Ce%5E%28a%5E2%29%2C%22%2C%220%3C=+x+%3C=+a%2C+e%5E%28x%5E2%29%2C+%22%2C%22+a+%3C+x+%3C+b%2C+e%5E%28b%5E2%29%2C+%22%2C%22b+%3C=+x+%3C=+c%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "This will ensure continuity at the separation points x = a and x = b. <BR>\n" );
document.write( "This will also make f(x) continuous over the closed interval [0, c], which by theorem makes it uniformly continuous.\r<BR>\n" );
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document.write( "NOTE:  As long as the continuity at the separation points are guaranteed, you may use any of the other choice functions, <BR>\n" );
document.write( "and the function f(x) will be uniformly continuous over the interval [0,c], by the (Heine-Cantor) theorem. \r<BR>\n" );
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document.write( "***By the way, you have a funny way of writing the sine function! </SPAN>\n" );
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