HOW TO DETERMINE IF A FUNCTION IS CONTINUOUS ON A GRAPH

About "How to Determine If a Function is Continuous on a Graph"

How to Determine If a Function is Continuous on a Graph :

Here we are going to see how to determine if a function is continuous on a graph.

Question 1 :

State how continuity is destroyed at x = xfor each of the following graphs.

(i)

Solution :

By observing the given graph, we come to know that

lim x-> x0- f(x)  =  f(x0)  (Because we have filled circle)

But,

lim x-> x0+ f(x)  ≠  f(x0)  (Because we have unfilled circle)

Hence the given function is not continuous at the point x = x0.

(ii)

Solution :

By observing the given graph, we come to know that

lim x-> x0- f(x)  =  f(x0)  (Because we have unfilled circle)

But,

lim x-> x0+ f(x)  =  f(x0)  (Because we have the same unfilled circle at the same place)

Hence the given function is continuous at the point x = x0.

(iii)

Solution :

From the given picture, we know that lim x-> x0- f(x)  =  -

But,

lim x-> x0- f(x)  = -

Hence it is not continuous at x = x0.

(iv)

Solution :

lim x-> x0- f(x)  =  f(x0)  (Because we have unfilled circle)

But,

lim x-> x0+ f(x)  ≠  f(x0)  (Because we have filled circle at different place)

Hence the given function is not continuous at the point x = x0.

Question 2 :

Consider the function f (x) = x sin π/x What value must we give f(0) in order to make the function continuous everywhere?

Solution :

f (x) = x sin π/x

Range of sin x is [-1, 1]

-1  ≤ sin π/x  ≤ 1

By multiplying x throught the equation, we get

-x  ≤ x (sin π/x)  ≤ x

Now let us apply the limit values

lim x -> 0 (-x)  ≤ lim x -> 0 x (sin π/x)  ≤ lim x -> 0 x

0 ≤ lim x -> 0 x (sin π/x)  ≤ 0

By sandwich theorem

lim x -> 0 x (sin π/x)  =  0

Now let us redefine the function

From this we come to know the value of f(0) must be 0,  in order to make the function continuous everywhere

Question 3 :

The function f(x)  =  (x2 - 1) / (x3 - 1) is not defined at x = 1. What value must we give f(1) inorder to make f(x) continuous at x = 1 ?

Solution :

By applying the limit value directly in the function, we get 0/0.

Now let us simplify f(x)

f(x)  =  (x2 - 1) / (x3 - 1)

=  (x + 1) (x - 1)/(x - 1)(x2 + x + 1)

=  (x + 1) / (x2 + x + 1)

lim x-> 1 f(x)   = lim x-> 1  (x + 1) / (x2 + x + 1)

=  (1 + 1)/ (1 + 1 + 1)

=  2/3

By redefining the function, we get

From this we come to know the value of f(1) must be 2/3,  in order to make the function continuous everywhere

After having gone through the stuff given above, we hope that the students would have understood, "How to Determine If a Function is Continuous on a Graph"

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