how can we find that a funtion is continuous?

Answer 1

A function f(x) is continuous at a point (x=a), only if:

It's Left-hand side limit equals it's Right-hand side limit.

Definition:

LHS limit at (x=a):

lim(h->0) f(a-h)

RHS limit at (x=a):

lim(h->0) f(a+h)

Only if these two limits are equal to each other, then the function f(x) is continuous at the given point (x=a)




Let me give you an easy example:

Suppose our function is f(x)=(1 / x)

We want to check whether this function is continuous at (x=0).

So, first we start by taking it's LHS limit:

lim (h->0) f(a-h)
= lim (h->0) f(0-h)
= lim (h->0) (1 / (0-h))
= lim (h->0) (-1/ h )

Now, keep this as it is & proceed to find the RHS limit:

lim (h->0) f(a+h)
= lim (h->0) f(0+h)
= lim (h->0) (1 / (0+h))
= lim (h->0) (1/ h )

Now, we compare our LHS and RHS limits, we find that both of them are almost the same - BUT, the LHS limit has a negative value & the RHS limit has a positive value. Hence, they are not completely equal.

Therefore, we say that our function f(x) = (1 / x) is NOT CONTINUOUS at (x=0).

Answer 2

A function is continuous in an domain if it is continuous in any point of that domain.
To verify that f is continuous at a point P, show that for every sequence Pn ---> P you have
f(Pn) ---> f(P) as n ---> infinity
Note that this is very general, and works for any function between two metric spaces, not only for one variable functions.

Practically, you generally have to show that your function is a composition of some "elementary" continuous function in their own definition's domain. The problematic point can be just in verifing which is the exact domains to work with (look for boundaries specifically).

Ex: to show that exp(sin(x-1)) is continuous on the entire real domain you have to simply notice that the exponential and the sine functions are continuous (and also that x ---> x-1 is continuous as well)

Answer 3

Usually, if there is any value of the variable that will cause division by zero, the function is not continuous at that value. In the example above, y = 1/x, you can see that the value x=0 will cause division by zero. The function is not continuous at that point.

If there is no value of the variable that will cause division by zero, the function is continuous.

Answer 4

A function is continuos at a point if
1) it is defined at a point
2) and left hand limit is right hand limit

for example

x+2 is continuous at each point.

tan x is not contiuous at pi/2 as left hand limit = inf right hand limit = - inf
you can refer below for proper understanding

Answer 5

A function is continuous if when your sketch it you do not lift your pen from the paper....(you do need to graph it)....if there are aymptotes, holes and boundaries then its not a continuous function

Answer 6

a function f(x) will be continuous at x=a if
1.f(a) exists
2.limit x>a of f(x) exists
3.f(a)=limit x>a of f(x)