What is Intermediate Value Theorem
Let's say you checked the continuity of the function of closed intervals and you found that the function is continuous. That is not all, you also need to check whether the function passes through the expected points. We call that expected point an Intermediate Value. For example, a function proves to be continuous on ending points, "a" and "b". A point, "c", is somewhere between both points. If the graph is continuous, it should pass through point "c" as well. To prove that, we use a theorem which is called the Intermediate Value Theorem. This theorem says, "If a function is continuous on the closed interval 
and k is any number between 
and 
then there exists a number, c, within 
such that 
."
By observing the graph, the intermediate value theorem can be defined another way:
- If a function is continuous on the closed interval , the function takes all values between and in this interval.
- The intermediate value theorem does not indicate the value or values of c, it only determines their existence.
You must be wondering why the intermediate value theorem is important? Because it provides valuable data. This theorem can help in the evaluation of the function at a specific point. Furthermore, it can be used to prove the existence of the roots of an equation.
Example
Prove that the image of 
exists in function 
.
The function is continuous in 
as it is the product of two continuous functions.
Take the interval 
, and study the value of the extremes:
Therefore there is a 
such that 
.
Prove that the image of 
exists in function 
with intervals 
.
The function is continuous in as it is the product of two continuous functions.
Studying extreme values:
Therefore there is a 
such that 
.
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Derivative of denominator is wrong
Good catch—thanks for pointing that out! We’ll double-check the derivative in that section and make any necessary corrections. Really appreciate you taking the time to flag it. 👍
Thank you
Thank you Abbas! Good luck with your studies!
With regard to the Zero Over a Number item, is there a mis-statement? It’s immediately followed by “If a number is divided by zero which means that the numerator is zero and the denominator is the number, then the result is zero.”
Hi Mark,
You’re absolutely right to raise the question — there does appear to be a misstatement in that sentence. The phrase “If a number is divided by zero, which means that the numerator is zero and the denominator is the number…” is indeed misleading and should be corrected.
To clarify:
Zero divided by a number (e.g. 0 ÷ 5) equals 0.
A number divided by zero (e.g. 5 ÷ 0) is undefined.
We’ll update the sentence to reflect the correct mathematical explanation. We appreciate you catching that and helping us improve the accuracy of the content!
There is more than one size of infinity, though. What if you multiply the infinity of the whole numbers (Aleph-0) by the infinity of the real numbers (fraktur-c)?
Thanks a lot to you for this essentiol article.
Hi Piyash! Thanks for your comment, great to hear that you found this useful!
Very nice