document.write( "Question 1145907: what do you consider that a function is continuous.How to see if it exists.
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\n" ); document.write( "Show by means of an example that limit as x approaches a[f(x)+g(x)] may exist even through neither limit as x approaches a f(x) nor limit as x approaches a g(x)exists.\r
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Algebra.Com's Answer #767191 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
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document.write( "Consider this function, defined on the number line for all real numbers\r\n" );
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document.write( "              / 0 if x is irrational,\r\n" );
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document.write( "    f(x) =  <\r\n" );
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document.write( "              \ 1 if x is rational.\r\n" );
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document.write( "This function is discontinued at every point on the number line.\r\n" );
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document.write( "Next consider function  g(x) = 1 - f(x).\r\n" );
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document.write( "This function is discontinued at every point on the number line, too.\r\n" );
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document.write( "But the sum of these two functions  f(x) +g(x) = f(x) + (1 - f(x)) == 1  is identically equal to 1 \r\n" );
document.write( "in all number line and is, therefore, continue function in all points of the number line.\r\n" );
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