CONTINUITY OF A FUNCTION

The function f is said to be continuous if f is a real function and c be the point on its domain of f then the left side limit is equal to the right-hand limit.  

If f is a real function on a subset of the real numbers and let c be a point in the domain of f. Then, f is continuous at c if 

                             xcf(x)=f(c)

CONTINUITY OF A FUNCTION-

Let us understand the continuity of function by example, then according to it, we will understand the definition of continuity.  

Let us consider  function defined as 

                              f(x)= {1, if x≠0

                                            {2, if  x=0

We can see that this function is defined at every point. Left and right hand limits at x=0 are both equal to 1. But the value of the function at x=0 equals 2 which does not coincide with the common value of the left and right hand limit. 

We can also see that we cannot draw the graph of the function without lifting the pen. This is yet another instance of a function being not continuous at x=0 . 

The definition of continuity of a function is given by-

If f is a real function on a subset of the real numbers and let c be a point in the domain of f. Then, f is continuous at c if 

                             xcf(x)=f(c)

Explanation- If the left hand limit , right hand limit and the value of the function at x=c exists and is equal to each other, then f is said to be continuous at x=c. If the right hand limit and left hand limit at x=c coincide, then we say that the common value is the limit of the function at x=c. 

In other words, the definition of continuity is- a function is continuous at x=c if the function is defined at x=c and if the value of the function at x=c equals the limit of the function at x=c. If f is not continuous at c, we say f is discontinuous at c and c is called the point of discontinuity of f. 

EXAMPLES OF CONTINUITY OF A FUNCTION-

1.  Check whether the function f(x)=x2 is continuous at x=0. 

     Sol- The function is defined at a given point x=0 and its value is 0.

              Then find the limit of the function at x=0.

                   x→0f(x) = x→0x2= 0

                   Thus, x→0f(x)=0= f(0)

                  Hence, f is continuous at x=0. 

1. Check the continuity of the function f given by f(x)=|x| at x=0.

      Sol- According to definition 

                                           f(x)= { -x, if x<0 

                                                   { x, if x ≥0

            The function is defined at 0 and f(0)=0. Left hand limit of f at 0 is

                             x0-f(x)=x0-(-x)= 0

           The right hand limit of f at 0 is  x0+f(x)=x0+x= 0

           Thus, the left hand limit, right hand limit and the value of the function coincide at x=0.                  

           Hence, f is continuous at x=0. 

HOW TO CHECK CONTINUITY OF A FUNCTION-

In order to find the continuity of a function, you need to check the Left hand limit( R.H. L.) and right hand limit ( R.H. L) of a function. After calculating the LHL and RHL of a function, check if they are equal. If the limits are equal then, the given function will be continuous and if not it is not continuous.  

DIFFERENTIABILITY-

The derivative of a real function is defined as –

Let f be the real function and c be the point on its domain. The derivative of f at c is given by

                      h→0f(c+h)- f(c)h provided this limit exists. 

Derivative of f at c is represented by f’(c). The function defined by

                              f’(x)  =h→0f(x+h)-f(x)h  wherever the limit exists is defined to be the derivative of f.

If a function f is differentiable at point c, then it will continuous at that point.

It is important to remember that every differentiable function is continuous. 

Let us go through some of the examples-

Find the dydx if y +siny = cos x

  Sol-  Differentiating the given function with respect to x, 

                dydx+ ddx( sin y)= ddx(cos x)

         Using chain rule,    dydx+ cos y.dydx= -sinx

                                        dydx=-sinx1+cos y where y≠ (2n +1)

ALGEBRA OF CONTINUOUS FUNCTION-

Since continuity of a function at a point is entirely related to the limit of the function at that point, then –

If f and g are two real functions continuous at a real number c. Then, 

1. f+g is continuous at x=c
2. f-g is continuous at x=c
3. f.g is continuous at  x=c
4. (fg) is continuous at x=c, provided g(c)≠0

Every rational function is continuous. 

CONCLUSION-

The function f is continuous –

If f is a real function on a subset of the real numbers and let c be a be point in the domain of f. Then, f is continuous at c if 

                             xcf(x)=f(c)

In order to check the continuity of a function, you have to check the right-hand limit and the left-hand limit separately. If both of the limits are the same, then the function will be continuous. We have also discussed that every differentiable function is continuous. A real-valued function is continuous at a point in the domain if the limit of the function at that point equals the value of the function at that point. A function is continuous if it is continuous on the whole of its domain.