Functions Limits and Continuity


   
 
Continuity at a Point
A function f (x) is said to be continuous at x = a if
 
f (a) exists.
 
 
 
Continuity in an Interval
A function f(x) is said to be continuous in an interval I, if it is continuous at each point of I.
 
 
then x can approach c both from the left and the right and so for f (x) to be continuous at c, we must have
 
 
If the interval I is the closed interval [a,b], then x cannot approach a from the left and it cannot approach b from the right. In this case f (x) is continuous at a, if it is continuous at b, if and it is continuous at
 
 
Example:
 
Discuss the continuity of the function f given by
 
f(x) = |x - 1| + |x - 2| at x=1
 
Suggested answer:
 
Right hand limit at x = 1
 
 
 
 
 
= h + (-h+1)
 
 
= 1
 
Left hand limit at x = 1
 
 
 
 
 
=1
 
f(1) = |1 - 1| + |1 - 2|
 
= 1
 
 
The function is continuous at x = 1.
 
Note 1:
 
We say that f(x) is continuous if f(x) is continuous at every point in its domain.
 
Note 2:
 
If f and g are two continuous functions then f + g, f - g, fg are continuous functions.
 
Note 3:
 
Every polynomial function is continuous.
 
Note 4:
 
Every rational function is continuous at each point of its domain.
 
Note 5:
 
Composition of two continuous functions is continuous.
 
Removal Discontinuity of a Function at a Point
 
If limit of a function f exists at a point c, but it is not equal to the value of the function at c.
 
 
But if f(c) l, then the function is said to have removal discontinuity at x = c.
 
This type of function can be made continuous by charging the value of f (c).
 
If we change f(c) = l, the function becomes continuous at x = c.
 
Example:
 
Show that the function
 
 
has a removal discontinuity at x = 4. Redefine the function f(x) at
 
x = 4 to make it continuous.
 
Suggested answer:
 
 
 
 
f (x) is not continuous at x = 4.
 
If we define f (x) = 256 at x = 4, then
 
 
Theorem 10:
 
If f and g are real functions such that fog is defined, if g is continuous at a point c, and if f is continuous at g(c), then fog is continuous at c.
 
Proof:
 
Since g is continuous at c, we have
 
 
Again, f is continuous at g(c) and so
 
 
 
 
 
 
 
Hence fog is continuous at c.
 
  
     
   
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