5.1 Introduction - Continuity

In this chapter we are going to learn the concepts of

1. continuity,

2. differentiability and

3. relations between both of them

4. differentiation of inverse trigonometric functions

5. exponential functions

6. logarithmic functions


Look at the following function and its geometrical represenation - 

begin mathsize 22px style f open parentheses x close parentheses equals open curly brackets table attributes columnalign left end attributes row cell 1 comma space x less or equal than 0 end cell row cell 2 comma space x greater than 0 end cell end table close end style

  function

 

Value of the function f(x) at x=0 and x<0 is 1. 

Value of the function f(x) at x>0 is 2. 

Left hand limit of f(x) at x=0 is 1 and right hand limit of f(x) at x=0 is 2.

Therefore  Left Hand Limit ≠  Right Hand Limit (or both limits do not coincide)

Value of the function at x=0 i.e.  f(0) = 1  { or it coincides with the Left Hand Limit }

We can't draw this graph in one stroke (without lifting the pen, at x=0 we need to lift the pen) 

Therefore this function f(x) is not continuous at x=0. 

Let us take another example -

Given below is a piecewise function and its geometrical representation

begin mathsize 22px style f open parentheses x close parentheses equals open curly brackets table attributes columnalign left end attributes row cell 1 comma space i f space x not equal to 0 end cell row cell 2 comma space i f space x equals 0 end cell end table close end style 



Conclusion:-  A function is continuous at particular point if it can be drawn without lifting the pen around that particular point. 

Definition 1 

Suppose 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 -

                                                  begin mathsize 18px style limit as x rightwards arrow c of f open parentheses x close parentheses equals f open parentheses c close parentheses end style


if the left hand limit = right hand limit = f(c)  {the value of the function at x = c } then f is said to be continuous at x = c.

Conclusion - 

A function is continuous at x = c if

  • it is defined at x = c and
  • the value of the function at x = c equals the limit of the function at x = c


Point of discontiuity - point where the given function is not continous. 



पिछ्ला सुधार: सोमवार, 23 अक्तूबर 2023, 5:05 अपराह्न