Concept of Continuity

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_(x-> c)` f(x) = f(c)
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. 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.

A real function f is said to be continuous if it is continuous at every point in the domain of f.
This definition requires a bit of elaboration. Suppose f is a function defined on a closed interval [a, b], then for f to be continuous, it needs to be continuous at every point in [a, b] including the end points a and b. Continuity of f at a means  
`lim_(x->a^+)` f(x) = f(a)
and continuity of f  at b means
`lim_(x->b^-)`  f(x) = f(b)
Observe  that `lim_(x->a^-)` f(x) and `lim_(x->b^+)` f(x) do not make sense. As a consequence of this definition, if f is defined only at one point, it is continuous there, i.e., if the domain of f is a singleton, f is a continuous function.