Continuity of Functions

  • Limit of a Function

Let f(x) be a function of a real variable x. We say that limit of f(x) as x \to a is L, when x assumes values closer and closer to a, f(x) assumes values closer and closer to L. We write this as,

{\lim \limits_{x \to a} f(x) = L}

  • Right Hand Limit and Left Hand Limit

x can approach a from 2 directions, either from its left [x < a] or from its right [x > a]. This gives rise to the right and left hand limits. Right hand limit is denoted by

{\lim \limits_{x \to 0^{+}} f(x)}

and left hand limit is denoted by

{\lim \limits_{x \to 0^{-}} f(x)}

Obviously, when both right and left hand limits are equal, \lim \limits_{x \to a} f(x) is defined.

  • Continuity and Discontinuity

The concept continuity can be regarded as the graphical viewpoint of limits. If we can draw the graph of a function f(x) without lifting our pen from the plane of paper, we say that the function is continuous everywhere. In other words, it does not have any gaps in its graph. When it has (at least one), we say that the function is discontinuous.

  • Continuity at a Point

When following equalities hold, function is continuous at x = a.

{\lim \limits_{x \to 0^{+}} f(x) = \lim \limits_{x \to 0^{-}} f(x) = f(a)}

This means, existence of limit is a NECESSARY but NOT SUFFICIENT condition for the function to be continuous. The limit of the function at a point a must be equal to the value of the function at that point, i.e. f(a).

  • Removable Discontinuity

The discontinuity of a function f(x) at a point a is said to be removable, if the limit as x \to a exists, but is not equal to f(a), or

{\lim \limits_{x \to 0^{+}} f(x) = \lim \limits_{x \to 0^{-}} f(x) \ne f(a)}

It can be removed by redefining the function at x = a.

  • Irremovable Discontinuity

The discontinuity of a function f(x) at a point a is said to be irremovable, if the limit as x \to a does not exist, or

{\lim \limits_{x \to 0^{+}} f(x) \ne \lim \limits_{x \to 0^{-}} f(x)}

This discontinuity cannot be removed and there is a jump in the graph of function.

  • Algebra of Continuity of Functions

Let f(x) and g(x) be 2 real valued functions of x \in \mathbb R. Suppose they are continuous at x = c. Then, following functions are continuous at x = c :

1) f(x) + g(x),

2) f(x) - g(x)

3) f(x) \times g(x)

4) f(x) \times k , k \in \mathbb R and

5) \frac {f(x)}{g(x)}, g(x = c) \ne 0

  • Continuity in an Interval

A function is said to be continuous in an interval, if it is continuous at each point in that interval.

  • Continuity in the Domain of a Function

A function is said to be continuous in its domain, if it is continuous at each point in the domain.

Recall that domain is that set of values, for which f(x) is defined.

  • Continuity of Few Functions

1) Polynomial functions, f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots + a_n x^n , n \in \mathbb N

They are continuous everywhere.

2) Rational functions

If f(x) and g(x) are polynomial functions, then \frac {f(x)}{g(x)} is continuous everywhere except at point/s where g(x) = 0.

3) Trigonometric functions

Sine and cosine functions are continuous everywhere.

Secant and tangent are continuous everywhere except at those points when cosine is 0.

Cosecant and cotangent are continuous everywhere except at those points when sine is 0.

4) Exponential function

f(x) = a^x , a > 0, a \ne 1 is continuous everywhere.

When a = 1, it becomes a constant function, which is also continuous everywhere.

5) Logarithmic function

f(x) = log_a x, a>0, a \ne 1 is continuous for every positive real number.

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