Functions Limits and Continuity

A function f (x) is said to be continuous at x = a ifA 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.