Categories: Math

# Continuity question

Question

Is this continuous?
How do i check if it is continuous or not?

g(x) = e^(1/x) for x < 0
Sin inverse (x) for 0 =< x =< 1
Tan inverse [1 / (x-1)] for x > 1

Its difficult to do “limits” and all by “typing”.
I’ll give you an idea.

A general rule of thumb to “check continuity” by “looking” is to see :

1) Whether the function is defined for all x. If its not, then it cant be continuous for a range that contains the “undefined value of x”.

2) If the function is in a “fraction” form, since one CANNOT divide by zero, so the denominator must not be zero.

The “proper” way of checking for continuity is to take limit from “both sides” that is ” Lower bound of the range for x to the point x” and “Upper bound of the range for x to point x”. If these “two limits” MATCH, then the function is continuous.

A trick is to “see” whether for every x, there is a corresponding value for f(x) or not. If it is, then the function is continuous!

For x<0 , g(x) = e^(1/x) has values for all x in that range. Thus its continuous. If you wanted to get a “rigorous proof”, you just need to show the “limit argument” mentioned earlier.

Let y = Sin inverse (x) for 0 =< x =< 1, since
sin y = x and for 0 =< x =< 1, we have corresponding values for y, means that y is continuous.

Let y = Tan inverse [1 / (x-1)] for x > 1,
Since Tan y = [1 / (x-1)] for x>1, does all values of y are “well defined”? yes. Thus y is continuous as well.

I have avoided the “rigorous method” which will require a lot of other “bridges of knowledge” but just wanted to point out how to understand what is meant by continuity and how to “see it ” in functions.

Tags: MATH
Anonymous Christian

Jonathan Ramachandran is founder at AnonymousChristian.org which serves as a Christian Blog discussing the Greater Hope of Biblical Christian in Christ Alone. He is also an IMO (International Math Coach) and a featuring Guitarist for many bands.

## Chiliasm Book 28 May 2023 Update with Martin Luther First Protestant in Last Chapter

Latest Update: 6 June 2023 I will just quote some major summary in images (more details in the free pdf… Read More

1 week ago

## Chiliasm Book Latest Update (10 New Chapters Added on 25 April 2023)

Latest Update: 28 May 2023 I will just quote some major summary in images (more details in the free pdf… Read More

1 month ago

## Did you know – First Christianity Chiliasm Possibility

Latest Update: 25 April 2023 I will just quote some major summary in images (more details in the free… Read More

3 months ago

## New Book titled Education for Daily Life

Latest Update: 22 February 2023 This post is mainly to announce the New Book titled "Education for Daily Life" which… Read More

4 months ago