Proving Existence of $\delta$ for Continuous Function $f$ s.t. $f(c)<5$

In summary, the conversation is about proving that for a continuous function f at a point c, if f(c) < 5, then there exists a delta value such that f(x) < 7 for all x in the interval (c - delta, c + delta). The conversation includes a discussion of how to find the appropriate delta value and how it relates to the given limit. The conclusion is reached that the proof can be simplified by choosing epsilon = 2 and using the given limit to show that the statement holds true. This proves that the delta value exists, as required.
  • #1
gimpy
28
0
Im having a little trouble with this question.

If f is continuous at [tex]c[/tex] and [tex]f(c) < 5[/tex], prove that there exists a [tex]\delta > 0[/tex] such that [tex]f(x) < 7[/tex] for all [tex]x \in (c - \delta , c + \delta)[/tex]

So we are given that f is continuous at c.
So [tex]\lim_{x \to c}f(x) = f(c) < 5[/tex]
[tex]\forall \epsilon > 0 \exists \delta > 0[/tex] such that whenever [tex]|x - c| < \delta[/tex] then [tex]|f(x) - f(c)| < \epsilon[/tex]

[tex]|x - c| < \delta[/tex]
[tex]-\delta < x - c < \delta[/tex]
[tex]c - \delta < x < c + \delta[/tex]

Ok now I am getting lost..
I know i have to do something with [tex]|f(x) - f(c)| < \epsilon[/tex]
maybe
[tex]|f(x) - 5| < \epsilon[/tex]
[tex]-\epsilon < f(x) - 5 < \epsilon[/tex]
[tex]5 - \epsilon < f(x) < 5 + \epsilon[/tex]
we want [tex]f(x) < 7[/tex] so ... am i on the right track? How can i find the [tex]\delta > 0[/tex] that satisfies this?
 
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  • #2
gimpy said:
[tex]5 - \epsilon < f(x) < 5 + \epsilon[/tex]
we want [tex]f(x) < 7[/tex] so ... am i on the right track? How can i find the [tex]\delta > 0[/tex] that satisfies this?

Yes, you're on the right track. Take [itex]\epsilon=2[/itex]. From the limit you mentioned, we know that there exists a [itex]\delta>0[/itex] such that [itex]3<f(x)<7[/itex] and so you are done. The question was asking you to show that a delta exists, and the limit demonstrates that it clearly does, so you are done.
 
  • #3
I was actually thinking about making [tex]\epsilon = 2[/tex] but thought it was to easy. Ok so thanks for explaining the question a bit more to me, i understand it now. :smile:
 
  • #4
gimpy,

Isn't what you're asked to prove equivalent to,

[tex]\lim_{x \to c}f(x) = f(c) < 7[/tex]

which follows trivially from,

[tex]\lim_{x \to c}f(x) = f(c) < 5[/tex]

which was what you were given?
 
  • #5
No, that's not what was given. Given that the limit is less than 5 it is true, but requires proof, that, for x close to c, f(x)< 5. gimpy was asked to prove the slightly simpler case: that, for x close to c, f(x)< 7.
 

What does it mean to prove the existence of $\delta$ for a continuous function?

Proving the existence of $\delta$ for a continuous function means showing that there exists a positive value of $\delta$ such that for any input value $x$ within a certain distance of a specific point $c$, the output value $f(x)$ will be within a certain range, such as less than 5.

Why is it important to prove the existence of $\delta$ for a continuous function?

Proving the existence of $\delta$ is important because it ensures that the function is continuous at the specific point $c$. This is essential for many mathematical and scientific applications, as it allows for the use of calculus and other tools to analyze the behavior of the function.

How do you determine the value of $\delta$ for a given continuous function?

The value of $\delta$ can be determined using the definition of continuity, which states that for any given $\epsilon > 0$, there exists a $\delta > 0$ such that if $|x-c| < \delta$, then $|f(x) - f(c)| < \epsilon$. This can be solved algebraically or graphically, depending on the function.

Can the value of $\delta$ be different for different points on a continuous function?

Yes, the value of $\delta$ can vary for different points on a continuous function. This is because the behavior of the function may be different at different points, so the distance needed to maintain continuity may also be different.

What happens if a continuous function does not have a value of $\delta$ that satisfies the given conditions?

If a continuous function does not have a value of $\delta$ that satisfies the given conditions, then it is not continuous at the specific point $c$. This means that the function has a discontinuity at that point, and the conditions of the problem cannot be met.

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