Continuity of Functions

continuous functions, evaluating functions, limits, value of a function

A continuous function is a function that does not have a break or a jump.  It is fairly straightforward to check for continuity when we are given a graph.  With regards to Algebraic Verification of Continuity, we have to check through the following agreement of Limit and the value of a function at the limit point.  The definition is as follows.

A function f(x) is continuous at the point "a" if

f(a)=lim   f(x)
        x->a

Example:

Given a random function f(x)=2x²+2

suppose we want to know if this function is continuous at the point x=2, then, we evaluate the function at this point.  That is:

f(2)=2(2)²+2=4+2=6  And we also find lim  f(x)=6
                                                                 x->2

This is so because the limit is the same thing as evaluating the function at that point.  so, we see that the limit agrees with the value of the function at the point "x=2"  Consequently, we say that this function is continuous at "x=2."

This seems redundant but it is not so.  There are functions that are not defined at a certain point or piecewise functions where the limit does not agree with the value of the function at a given point.  Analyzing these sorts of functions allows for a better understanding of continuity of functions

continuous functions, evaluating functions, limits, value of a function