Continuity at a Point via Pictures
The Pencil Rule of Continuity
A continuous function is one that we can draw without lifting our pencil, pen, or Crayola crayon.
Here are some examples of continuous functions:
If a function is continuous at x = c we can start with our pencil a little to the left of x = c and trace the graph until our pencil is a little to the right of x = c, without lifting our pencil along the way.
We will now return to those functions that are continuous at x = c. We can trace each function with a pencil, from one side of x = c to the other, without lifting the pencil.
If a function isn't continuous at x = c, we say it's discontinuous at x = c.
This function is not continuous at x = c, since the function isn't even defined at x = c. We can't compare the value of f(c) to , since neither exists!
This function "jumps" at x = c. To draw the graph we would have to draw one line, stop at x = c and lift the pencil, then draw another line. As far as the limit definition goes, doesn't even exist (the one-sided limits disagree). Therefore f can't possibly be continuous at c.
This function also jumps at x = c. To draw the graph we would have to draw a line, lift the pencil and draw a dot at x = c, then lift the pencil again to draw the remaining line. In this graph both f(c) and exist, but the function value disagrees with the limit.
If a function f is discontinuous at x = c, then at least one of three things need to go wrong. Either
- f(c) is undefined (therefore we can't draw it at all),
- we need to move the pencil either just before or just after we reach x = c ( doesn't exist), or
- we need to move the pencil either just before or just after we reach x = c and we need to draw a separate little dot for f(c).
In other words: for a function f(x) to be continuous at x = c, three things need to happen:
- The function must be defined at x = c (that is, f(c) must exist)
- The limit must exist (both one-sided limits must exist and agree)
- The value f(c) must agree with the limit .