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Justify if continuity can be affirmed, denied or cannot be decided in the point

x = 0

of a real function

f(x)

of real variable, in each of the three (independent) cases .
a) It is known only that for all natural

n

f\left( \frac{1}{2n}\right)= 1

and

f\left( \frac{1}{2n+1}\right)= -1

.
b) It is known that for all nonnegative real

x

f(x) = x^2

and for negative real

x

f(x) = 0

.
c) It is only known that for all natural

n

it is

f\left( \frac{1}{n}\right)= 1

Problem Statement