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

Answer

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.