 |
| Continuity at a Point |
|
| A function f (x) is said to be continuous at x = a if |
| |
 f (a) exists. |
| |
 |
| |
 |
| |
|
| 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. |
| |
|
| |
| 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. |
| |
|
| |
| Since g is continuous at c, we have |
| |
 |
| |
| Again, f is continuous at g(c) and so |
| |
| |
| |
 |
| |
 |
| |
 |
| |
 |
| |
 |
| |
| Hence fog is continuous at c. |
| |
