How can you tell if a function has a discontinuity?

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How can you tell if a function has a discontinuity?

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Answer 1 · 2015-06-19T22:50:23

Let be $f (x) : A \to B$ $f(x):A->B$ . The function is discontinuous in $RR$ if and only if $AsubRR$ and $A!=RR$ .
Pragmatically we can say that if a function has a fractional part, a logarithm, an even root, a tan(x), a cot(x), an arcsin(x), an arccos(x) there might be one or more points of discontinuity. The only way is to verify if there are points where the function isn't defined.
I make some examples:

$a(x)={(x if x>2),(2" if " 2>=x>1) :}$
$b(x)=log_5(3x)$
$c(x)=x/x$
$d(x)=arccos(x)$
$e(x)=e^x/3$
$f(x)=sqrt(abs(x))$

$D_"a(x)"={x|x in RR ^^x>1}$
$D_"b(x)"={x|x in RR ^^x>0}$
$D_"c(x)"={x|x in RR ^^x!=0}$
$D_"d(x)"={x|x in RR ^^0<=x<=pi}$
$D_"e(x)"=RR$
$D_"f(x)"=RR$
So e(x) and f(x) are continuous.

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How can you tell if a function has a discontinuity?

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