a) At x=3 the function is not defined (approaches infinity because of division by zero).
b) x²+x-6=(x+3)(x-2) so the function becomes x+3, except for x=2, when the function is undefined (hole) so it’s discontinuous at x=2.
c) The function is only defined when cos(x) is zero or positive, because square root of a negative number is a complex number (not real). The range of x is 0-90°, and then the function is undefined for 90°<x<270° and then defined for 270°-360°, the discontinuity continuing like this for every cycle.
d) INT is a “jumpy” function, since it can only generate integers. For example, for 0≤x<1, it is zero; for 1≤x<√2 it is 1, then for √2≤x<√3 it is 2. The function’s graph looks like a rising series of steps with no join between each level—discontinuity.
e) Not defined when x=0, 180°, 360°, etc., because sin(x)=0 at these points and the function has hole discontinuities. Near the holes the function has a value close to 1.
f) Not defined when x=0 where there is a hole. Function is 1 otherwise.
g) Since x² is always positive, except when x=0, the function is undefined (approaches -∞) when x=0.
h) The denominator is (x-3)² so is zero when x=3, therefore the function is undefined (approaches infinity).
i) Not defined for multiples of 90° when it approaches plus or minus infinity.