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What is a Continuous Function?

What is a continuous function?

Summary

  1. 'Disc.' is an abbreviation for 'discontinuous'
  2. Discontinuous functions have a jump or a break in their graphs
  3. Graph B has a break between x=2 and x=3, so it is discontinuous
  4. 'Cont.' is an abbreviation for 'continuous'
  5. Continuous functions have no jumps or breaks in their graphs
  6. Graph A has no jumps or breaks, so it is continuous
  7. We can trace Graph C without ever picking up our finger, so it must be continuous
  8. There is a break in Graph D at the y-axis, so it is discontinuous
  9. Graph E has two jumps, so it is discontinuous

Notes

    1. These two graphs both look like lines, but there is something different about them!
    1. Graph A has a y-value for every x-value on the graph
    2. Graph B has a gap in it -- there are no y-values for the x-values between 2 and 3
    1. Discontinuous functions have a jump or a break in them
    2. Graph B has a break in it between x=2 and x=3, so it is discontinuous
    1. Graph A has no jumps or breaks, which means it is continuous
    1. If the graph has any x-values that don't have y-values to go with them, then the function is discontinuous
    2. Graphs of discontinuous functions might also jump from one point to another at a particular x-value, making a break in the graph
    3. These functions are discontinuous as well
    1. Does Graph C have any breaks or jumps?
    1. This is a handy little trick to test whether or not a function is continuous
    1. If you have to pick up your finger, that means that there is some sort of break in the graph
    2. And if there is a break in the graph, the function is discontinuous!
    1. Is Graph D continuous or discontinuous?
    1. When you get close to the y-axis, the graph starts to go down forever and ever
    2. Then the part of the graph on the other side of the y-axis starts way up near positive infinity
    3. To get from one part of the graph to the next, we need to pick up our finger and 'jump' over the y-axis
    1. There is a break in Graph D at the y-axis
    2. It goes from negative infinity to the left of the y-axis and starts from positive infinity on the right
    3. The imaginary line going over the y-axis where the break occurs is called a vertical 'asymptote'
    4. An asymptote is a line that a graph approaches but never crosses and is another kind of discontinuity
    1. Is Graph E continuous or discontinuous?
    1. This graph has two spots where it jumps up from one point to another point nowhere near the first one
    2. Since we have jumps, this graph is discontinuous
    3. This graph represents a special type of function called a 'step function'
    1. Calculus uses a concept called a 'limit' to define continuous functions more specifically
    2. But you don't need to worry about that yet!
    1. For now, we can just look at the graph to see whether or not it's continuous