Boundedness

Key Questions

• What is boundedness?

  Answer:

  Boundedness is about having finite limits. In the context of values of functions, we say that a function has an upper bound if the value does not exceed a certain upper limit. More...

  Explanation:

  Other terms used are "bounded above" or "bounded below".

  For example, the function `f(x) = 1/(1+x^2)` is bounded above by `1` and below by `0` in that:

  `0 < f(x) <= 1` for all `x in RR`

  graph{1/(1+x^2) [-5, 5, -2.5, 2.5]}

  The function `exp:x -> e^x` is bounded below by `0` (or you can say has `0` as a lower bound), but is not bounded above.

  `0 < e^x < oo` for all `x in RR`

  graph{e^x [-5.194, 4.806, -0.74, 4.26]}

  George C. · 3 · Jun 29 2015
• What is the boundedness theorem?

  Answer:

  A continuous function defined on a closed interval has an upper (and lower) bound.

  Explanation:

  Probably the simplest boundedness theorem states that a continuous function defined on a closed interval has an upper (and lower) bound.

  Proof by contradiction

  Suppose `f(x)` is defined and continuous on a closed interval `[a, b]`, but has no upper bound.

  Then:

  `AA n in NN, EE x_n in [a, b] : f(x_n) > n`

  Since the sequence of `x_n`'s lies in a bounded interval, it is dense at some point in the closure of the interval. Since the interval is closed, that must be at some point `c` actually in the interval `[a, b]`.

  Since the sequence of `x_n`'s is dense at `c`, there is some monotonically increasing sequence `n_k in NN` such that `x_(n_k) -> c` as `k->oo`.

  Now `f(x)` is continuous at `c`, so:

  `lim_(x->c) f(x) = f(c)`

  which is bounded.

  But:

  `lim_(k->oo) x_(n_k) = c" "` and `" "lim_(k->oo) f(x_(n_k)) = oo`

  is unbounded.

  ...contradiction.

  So there is no such `f(x)` lacking upper (or lower) bound.

  George C. · 1 · Oct 14 2017
• How does the boundedness of a function relate to its graph?

  If the function is unbounded, the graph would progress to infinity, in some direction(s).

  A. S. Adikesavan · 1 · Mar 2 2016
• Are all functions bounded?

  Not all functions are bounded.

  The simplest counter example would be
  the identity function `f(x) = x`
  which is defined for all values of `x` and can generate any value for `f(x)`

  A slightly less trivial counter example would be
  the cubing function `f(x) = x^3`

  Alan P. · 1 · Apr 27 2015

• What is boundedness?
• What are some examples of bounded functions?
• Are all functions bounded?
• How does the boundedness of a function relate to its graph?
• What are some examples of unbounded functions?
• Is `y = 5` an upper bound for `f(x) = x^2 + 5`?
• What is the boundedness theorem?
• Is there a lower bound for `f(x) = 5 - 1/(x^2)`?
• Can an absolute value have a discontinuity `f(x)= |x-9| / (x-9)`?
• Suppose `I` is an interval and function `f:I->R` and `x in I` . Is it true that `f(x)=1/x` is not bounded function for `I=(0,1)` ?. How do we prove that ?