Continuity

Continuity Definition A function is said to be continuous in a given interval if there is no break in the graph of the function in the entire interval range. Assume that “f” be a real function on a subset of the real numbers and “c” be a point in the domain of f. Then f is continuous at c if x tends to a then f(x) tends to f(a)

In other words, if the left-hand limit, right-hand limit and the value of the function at x = c exist and are equal to each other, i.e.,

then f is said to be continuous at x = c Types of Discontinuity The four different types of discontinuities are:

Removable Discontinuity Jump Discontinuity Infinite Discontinuity

Conditions for Continuity A function “f” is said to be continuous in an open interval (a, b) if it is continuous at every point in this interval. A function “f” is said to be continuous in a closed interval [a, b] if f is continuous in (a, b)

Definition:  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

lim  f(x)= f (c )

x →c

More elaborately, if the left hand limit, right hand limit and the value of the function

at x = c exist and equal to each other, then f is said to be continuous at x = c. Recall that if the right hand and left hand limits at x = c coincide, then we say that the common value is the limit of the function at x = c. Hence we may also rephrase the definition of continuity as follows: a function is continuous at x = c if the function is defined at x = c and if the value of the function at x = c equals the limit of the function at x = c. If f is not continuous at c, we say f is discontinuous at c and c is called a point

of discontinuity of f.