Example:
If f(x) = x if x is rational and f(x) = 0 if x is irrational,
prove that x is continuous at 0.
If one looks at this poor
representation of the function, we see that it does not at all
look continuous. But if { 
}
is any sequence of numbers (rational or irrational) that converges
to zero, then there exists an integer N such that | |
< 
for n >
N. But f() is
either zero or itself,
and in any case we have
That proves that the sequence of { f()
} converges to 0 = f(0), which proves that the function is continuous
at zero.
As an exercise, prove that the function is not continuous for
any other x.
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