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Solved Examples on Continuity

Illustration 1: The function

f(x) = [log(1+ax) – log(1-bx)]/ x 

is not defined at x = 0. The value which should be assigned to f at x=0 so that it is continuous at 0 is (1983)

1. a-b                                                              2. a+b

3. log a + log b                                                 4. None of these

Solution: We know for the function f to be continuous we must have

f(0) = limx0 f(x)

       = limx0[log(1+ax) – log(1-bx)]/ x 

       = limx0a[log(1+ax)] /ax+ b log(1-bx)]/ -bx

       = a.1 + b.1 

       = a+b (since f(0) = a + b)

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Illustration 2: The function f(x) = [x]2 – [x2] is discontinuous at (1999)

1. all integers

2. all integers except 0 and 1

3. all integers except 0

4. all integers except 1

Solution: This question is based on greatest integer function. We know that all integers are critical points for the greatest integer function.

Case 1: When x ∈ I

f(x) = [x]2 – [x2] = x2 –x2 = 0

Case 2: when x does not belong to I

If 0<x<1, then [x] = 0

And 0<x2<1, then [x2] = 0

Next, if 1 ≤ x2< 2

Then 1≤ x < √2

So, [x] = 1 and [x2] = 1

Therefore, f(x) = [x]2 – [x2] = 0 if 1 ≤ x <√2

Hence, f(x) = 0 if 0≤ x < √2

This shows that f(x) is continuous at x =1.

Hence f(x) is discontinuous in (-∞, 0) ∪[√2, ∞) on many other points. 

Illustration 3: let f(x) = x+a, if x < 0

                               = |x-1|, if x ≥ 0

and g(x) = x+1, if x < 0

             = (x-1)2 + b, if x ≥ 0

Where a and b are non-negative real numbers. Determine the composite function gof....

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