Continuity- Introduction

A real valued function is continuous at a point in its domain if the limit of the function at that equals the value of the function at that point.

Definition 1:-   Suppose f be a real function on a subset of the real numbers and let ‘a’ be a point in the domain of ‘f’.  Then f is continuous at ‘a’ if

                               lim  f(x)  = f(a)
                               x — a

Definition 2:-    A real valued function is said to be continuous if it is continuous at every point in the domain of f
A function is said to be continuous at x=a, if

                            lim   f(x) =  lim  f(x)=  f(a)
                              x—a^-         x — a⁺

Domain of a function

Let f:A — B be a function then the set of first components in the ordered pair of the function is said to be the domain. In other words, first set A is the domain of the function.  B is called the co-domain of the function.

•  For example:  The domain of the modulus function, f(x)=|x| is R

                     The domain of the greatest integer function is also R

Real Valued Function

A function which has either R or one of its subsets as its range is called real valued function.  Further, if its domain is also either R or a subset of R, it is called a real function.

Discontinuous function

A function which is not continuous is called discontinuous function.

For a discontinuous function,   limf(x)  ≠  f(a)
                                          x—a

Graph of a discontinuous functions:

Algebra of continuous function

Theorem 1:-

  Suppose f and g be two real functions continuous at a real number ‘c’, then
1)  f + g is continuous at c
2)  f - g is continuous at c
3)  f. g is continuous at c
 

Theorem 2:-

Suppose f and g are real valued functions such that (f o g) is defined at c.  If g is continuous at c and if f is continuous at g( c), then       (f o g) is continuous at c.

For example: Let f(x) = sin(x²)

Take g(x) = sinx and h(x)=x², both the functions are continuous, so that g o h = g[h(x)]=sin(x²) is also continuous.

Example 2: Find all the points of discontinuity of the function f defined by
                        

Solution: Left hand limit,  limf(x) = lim x+2 = 1+2=3
                                x — 1^-   x — 1^-

             Right hand limit,  limf(x) = lim x-2 = 1-2=-1
                                   x — 1⁺     x — 1⁺

Since, the left hand limit is not equal to right hand limit at x=1, the only point of discontinuity is x=1.
                        


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Reference Links:

•  http://en.wikipedia.org/wiki/Continuous_function
•  http://en.wikipedia.org/wiki/Subset
•  http://en.wikibooks.org/wiki/Algebra/Functions
•  http://en.wikipedia.org/wiki/Domain_of_a_function
•  http://www.wordiq.com/definition/Greatest_integer_function