Could you explain to me in a very simple way that I can understand the definition of continuity ?

Question

It's very hard for me to understand this and it doesn't seem logical to me. This is from Calculus by the way.

Ok it says that a function f(x) is continuous at a point x=c if the following three conditions are met:

This is the part that I don't understand, how do I solve this?

a) lim f(x), x-> c exists

b) f(c) is defined

c) lim f(x)= f(c) x->c

It seems so simple but to me is not. I am given an exercise about finding the continuity and I don't know with what I have to substitute "c" and "x" in those formulas.

At least explain to me what are those formulas for.

Thank You Very Much

Answer 1

What this means is that a function is continuous at a point c if it has a value at the point, and if the function approaches the value as x approaches c.

Imagine a parabola sitting on the x axis so that the value of the function at x=0 is zero. The function is f(x) = x²

f(0) = 0. This satisfies condition b.

The limit as x approaches 0 exists. This is true for a parabola.

Finally, as you approach x=0 along the parabola from either side, the value of the function approaches 0. So the limit = 0. If the limit equals the value, the the function is continuous at that point.

Now consider the function:

f(x) = x² when x not 0, f(x) = 1 when x=0.

This one has a value at x=0. It also has a limit as x approaches 0. The limit is 0. But the value of the function at x=0 is not equal to the limit as x approaches 0.

One final function:

f(x) = +1 when x is >=0; f(x) = -1 when x is <0

This has a value at x=0. The value is f(0) = +1.

But it deosn't have a limit, because when you approach 0 from the positive side, the value is 1 and when you approach it from the negative side, the value is -1. There is no limit, so the function is not continuous.

Hope this helps.

Answer 2

The continuity in calculus comes for graph.It means if one can draw a graph without lifting the pencil that means the graph is continuous.
Sometimes the condition of drawing graph is such that at some point one must lift the pencil to draw the whole graph.At that point the graph is not continous.
The meaning of x->c means x tends to c.It includes from both sides means first the value of x is greater than c and approaching towards c means deminishing. The reverse is also there means the value of x is lesser than c and approaching towards it, means increasing.
So if the two values and also the value of the function at that point means f(c) are equal.. the graph will be continuous.
Mathemeatically, you must be very clear about limit concept otherwise you must be very good in drawing graph pictorially.. so you can find the function is continuous or not at any point.

Answer 3

It might be an idea to make a rough sketch of the parabola y = f(x) = x² to discuss these conditions.
f(x) = x² is a continuous function in that there are no breaks.

From the graph:-
a) if x = c is a value of x on the x axis, then
lim as x -->c of f(x) = c²
i.e. lim as x--> c of f(x) exists.

b) f(c) = c²
i.e. f(c) is defined.

c) lim f(x) = f(c) as x ---> c

Answer 4

Continuous means there are no "breaks" in the graph. We translate this in mathematical terms by saying:

"f(x) gets closer and closer to f(c) as x gets closer and closer to c". But of course we mathematicians hate long sentences so we just write it shorthand:

lim f(x) = f(c).
x→c