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# Can a bounded function have an unbounded derivative?

## Can a bounded function have an unbounded derivative?

I think the answer is “yes”. If the graph were to have an unbounded derivative, it would coincide with a vertical line.

How do you determine if a function is bounded or unbounded?

A function that is not bounded is said to be unbounded. If f is real-valued and f(x) ≤ A for all x in X, then the function is said to be bounded (from) above by A. If f(x) ≥ B for all x in X, then the function is said to be bounded (from) below by B.

### What is bounded unbounded?

Generally, and by definition, things that are bounded can not be infinite. A bounded anything has to be able to be contained along some parameters. Unbounded means the opposite, that it cannot be contained without having a maximum or minimum of infinity.

What does it mean if a derivative is bounded?

Show activity on this post. It means that there exist constants m and M such that f′(x) always satisfies m≤f′(x)≤M.

#### Can a differentiable function be unbounded?

On the interval (−1,1), g(x) is bounded by 2. However, for ak=1√kπ with k∈N we have h(ak)=2√kπ(−1)k which is unbounded while limk→∞ak=0.

What is the boundedness theorem?

The boundedness theorem says that if a function f(x) is continuous on a closed interval [a,b], then it is bounded on that interval: namely, there exists a constant N such that f(x) has size (absolute value) at most N for all x in [a,b].

## Is Lnx bounded?

Bookmark this question. Show activity on this post. For 1≤x<∞, we know lnx can be bounded as following: lnx≤x−1√x.

What is bounded and unbounded sequence?

A sequence an is bounded below if there exists a real number M such that. M≤an. for all positive integers n. A sequence an is a bounded sequence if it is bounded above and bounded below. If a sequence is not bounded, it is an unbounded sequence.

### Is bounded function differentiable?

Differentiable functions being continuous, are bounded over S, if S is a closed and bounded subset of the domain of the function. Examples :— The function f(x) = x^2 is differentiable everywhere, but not bounded over R, whereas g(x) = sin(x) is bounded over R.

How do you prove a function is bounded?

Equivalently, a function f is bounded if there is a number h such that for all x from the domain D( f ) one has -h ≤ f (x) ≤ h, that is, | f (x)| ≤ h. Being bounded from above means that there is a horizontal line such that the graph of the function lies below this line.

#### What is unbounded function?

Now, a function which is not bounded from above or below by a finite limit is called an unbounded function. For example: – x is an unbounded function as it extends from −∞ to ∞. Similarly, tanx defined for all real x except for x∈(2n+1)π2 is an unbounded function.

What makes a function bounded?

A function f(x) is bounded if there are numbers m and M such that m≤f(x)≤M for all x . In other words, there are horizontal lines the graph of y=f(x) never gets above or below.