Question

How can you tell if a function has a discontinuity?

Answer 1

Let be `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.