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.