Continuity of function?

umerata

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find the points at which the function f(x) in the figure given below is continuous and the points at which f(x) is discontinuous

Graph:
Untitled-1.jpg

[FONT=arial, helvetica, clean, sans-serif]Please explain me how to do this question [/FONT]
[FONT=arial, helvetica, clean, sans-serif]Explain !![/FONT]

also explain left hand and right hand limit please
Thank you.
 
find the points at which the function f(x) in the figure given below is continuous and the points at which f(x) is discontinuous

Graph:
View attachment 2527

Please explain me how to do this question
Explain !!

also explain left hand and right hand limit please
Thank you.

What does your text-book say about "continuity of a function"?

What does your text-book say about "left hand and right hand limits"?

and...

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find the points at which the function f(x) in the figure given below is continuous and the points at which f(x) is discontinuous

Graph:
View attachment 2527

Please explain me how to do this question
Explain !!

also explain left hand and right hand limit please
Thank you.
You are taking a calculus course, correct? In English? Is English your native language?

There is a definition of the word "continuous." That definition should be in your course materials; if not, it is at wikipedia. Defined functionally, three conditions are required for a function to be continuous over a domain. What are those conditions?

There is a convention of notation about intervals on the real number line involving square brackets and parentheses. What is it? To be concrete, what intervals are described by [1, 2], [1, 2), (1, 2], and (1, 2), do you know?

There is a convention of graphing about circles. Do you know what solid circles mean? Do you know what empty circles mean?

Once you answer the questions above, I think you will see how to answer your first question on your own, but if not, someone will then be able to explain it to you easily.

\(\displaystyle \displaystyle \lim_{x \rightarrow a}f(x) = b\) informally means that when x is very close to a but not equal to a, f(x) either equals b or is very close to b. Do you fully understand this informal definition? What is informal about it?

The rigorous definition is

\(\displaystyle \displaystyle \lim_{x \rightarrow a}f(x) = b\ MEANS\ \exists\ \delta > 0\ such\ that\ 0 < |x - a| < \delta \implies |f(x) - b| < \epsilon\ for\ any\ \epsilon > 0.\)

Do you understand this rigorous definition and how it makes rigorous the informal definition given above?

If so, it is easy to explain right-hand and left-hand limits.
 
Your graph, by the way, does not make sense because it gives two values for the function at x= 4.
 
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