**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 = x_{0 }for each of the following graphs.

(i)

**Solution :**

By observing the given graph, we come to know that

lim_{ x-> x0- }f(x) = f(x_{0}) (Because we have filled circle)

But,

lim_{ x-> x0+ }f(x) ≠ f(x_{0}) (Because we have unfilled circle)

Hence the given function is not continuous at the point x = x_{0}.

(ii)

**Solution :**

By observing the given graph, we come to know that

lim_{ x-> x0- }f(x) = f(x_{0}) (Because we have unfilled circle)

But,

lim_{ x-> x0+ }f(x) = f(x_{0}) (Because we have the same unfilled circle at the same place)

Hence the given function is continuous at the point x = x_{0}.

(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 = x_{0.}

(iv)

**Solution :**

lim_{ x-> x0- }f(x) = f(x_{0}) (Because we have unfilled circle)

But,

lim_{ x-> x0+ }f(x) ≠ f(x_{0}) (Because we have filled circle at different place)

Hence the given function is not continuous at the point x = x_{0}.

**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) = (x^{2} - 1) / (x^{3} - 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) = (x^{2} - 1) / (x^{3} - 1)

= (x + 1) (x - 1)/(x - 1)(x^{2} + x + 1)

= (x + 1) / (x^{2} + x + 1)

lim _{x-> 1} f(x) = lim _{x-> 1} (x + 1) / (x^{2} + 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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