Continuity on an Interval via Pictures
Remember, f is continuous on an interval if we can finger paint over f on that interval without lifting our drawing digit.
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).