Continuity: Learn It 3

Continuity Over an Interval

Now that we have explored the concept of continuity at a point, let’s extend it to continuity over an interval. A function is continuous over an interval if you can trace it without lifting your pencil between any two points within that interval.

  • Open Interval: A function is continuous on an open interval (a,b) if it is continuous at every point within that interval.
  • Closed Interval: A function is continuous on a closed interval [a,b] if it is continuous on (a,b), continuous from the right at a, and continuous from the left at b.

continuity over an interval

  • A function is continuous over an open interval if it is continuous at every point in the interval.
  • A function f(x) is continuous over a closed interval of the form [a,b] if it is continuous at every point in (a,b) and is continuous from the right at a and is continuous from the left at b.
  • Analogously, a function f(x) is continuous over an interval of the form (a,b] if it is continuous over (a,b) and is continuous from the left at b.

Requiring that limxa+f(x)=f(a) and limxbf(x)=f(b) ensures that we can trace the graph of the function from the point (a,f(a)) to the point (b,f(b)) without lifting the pencil. If, for example, limxa+f(x)f(a), we would need to lift our pencil to jump from f(a) to the graph of the rest of the function over (a,b].

How To: Determine Continuity Over an Interval

  1. Check Continuity on Open Interval: Verify the function is continuous at all points within (a,b).
  2. Check Right Continuity at  a: Ensure limxa+f(x)=f(a).
  3. Check Left Continuity at b: Ensure limxbf(x)=f(b).

State the interval(s) over which the function f(x)=x1x2+2x is continuous.

State the interval(s) over which the function f(x)=4x2 is continuous.