- Topics At a Glance
- Continuity at a Point
- Continuity at a Point via Pictures
- Continuity at a Point via Formulas
- Functions and Combinations of Functions
- Continuity on an Interval
- Continuity on an Interval via Pictures
- Continuity on an Interval via Formulas
- Continuity on Closed and Half-Closed Intervals
- Determining Continuity
- The Informal Version
- The Formal Version
- Properties of Continuous Functions
- Boundedness
- Extreme Value Theorem
- Intermediate Value Theorem
- In the Real World
- Page: I Like Abstract Stuff; Why Should I Care?
- Page: How to Solve a Math Problem
- Appendix: Intervals and Interval Notation
Introduction to :
Remember, f is continuous on an interval if we can finger paint over f on that interval without lifting our drawing digit.
Sample Problem
Look at the function f drawn below:
- The function f is continuous on the interval (-5,5) because if c is any point in (-5,5),
. We can start our pencil out on the graph at x = -5 and trace the graph to x = 5 without lifting the pencil. - This function is not continuous on the interval (5,8) because f is not continuous at x = 7. When x = 7 we need to lift the pencil to trace the graph.
- Here's a tricky one: the function f is continuous on the interval (5,7). The only point on the whole graph at which f is discontinuous is x = 7, and 7 isn't in the interval (5,7).
Practice:
Look at the function f drawn below:
Determine whether the function is continuous on the given interval. If not, state the points in the interval at which f is discontinuous.
- (-5,-3)
Determine whether the function is continuous on the given interval. If not, state the points in the interval at which f is discontinuous.
- (-5,3)
Determine whether the function is continuous on the given interval. If not, state the points in the interval at which f is discontinuous.
- (9,10)
Determine whether the function is continuous on the given interval. If not, state the points in the interval at which f is discontinuous.
- (-3,-1)
Determine whether the function is continuous on the given interval. If not, state the points in the interval at which f is discontinuous.
- (0,4)
