Find at least one point at which each function is not continuous and state which of the 3 conditions in the definition of continuity is violated at that point.

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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.

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