Limits and Continuity

In the last topic we, we worked with some limit theorems to evaluate limits of different functions. In this section, we are going to discuss continuity which is an important concept in Calculus. Without continuity, we are not able to differentiate which is half of Calculus.

The idea behind continuity is like it's name suggests, the function follows a path that is not broken. It's kind of like drawing a line without lifting up your pencil.

That's the idea of continuity which is not a careful definition of continuity. In order for a function to be continuous at a given point c, the following limit must be true:

Using the espilon/delta definition of a limit we can say a function,, is continuous at c if and only if:

For all > 0, there exists a delta > 0, such that if 0 <|x - c|< delta, then |f(x)-f(c)|<epsilon.

This is kind of like the epsilon/delta definition of a limit, but this time L is replaced by f(c).

The final concept in the theory of continuity is the definition of left-continuous and right-continuous. A function is called left-continuous or continuous from the left if and only if:

A function is said to be right-continuous or continuous from the right if and only if:

That's about it for definitions. There's also a theorem that says if f and g are contiuous at c then, f+g,f-g,fg, and f/g g0 are also continuous at c. That's about it for theory. Let's work some problems.

Suppose we have a function that is defined:

Is the function continous at x = 5? Graphing the function sometimes helps resolve any discontinuities:

The function is not continuous at x = 5 because:

but we can say this about the function the way it is now:

Even though there is a discontinuity with , it is easy to fix. Simply let = 5 at x = 5. Adding the point (5,5) makes the function continuous. The discontinuity was classified as removable