Mathematical Continuity

The concept of limits and continuity was developed by various mathematicians in about 17th century with a key aim of developing calculus. Although implied in the development of calculus of the 17th and 18th centuries, the modern concept of limit and continuity backdates to Bolzano who, in 1817, introduced the fundamentals of the epsilon-delta technique to explain functions that are continuous. There is no clear source since the works of Bolzano were not famous during his lifetime.  In 1821, Cauchy explained limits in his Cours d’analyse and produced the modern information, which is not mostly used since it is a verbal definition. It was Weierstrass who first introduced the epsilon-delta definition of limit as is is today used.  He is also the one who introduced the symbols limx→x0 and lim. Calculus is an important branch of mathematics that is focused on limits, functions, integrals, derivatives, and infinite series. a function is said to be continuous if a small change in the input produces an equally significant change in the results. If a function, say f(x) gets nearer and nearer to p, as x approaches q, then the limit of f, at q, is p. Then, If f (q) = p, it Implies that f is continuous at p. By instinct, a function that is continuous can be easily graphed. The criterion called “close to", which will be better understood clearly in this text, depends on the distance concept which is more sensible in defining distance in three dimensions which has three axis as shown in a figure below. The generalization of the three dimension space came between the 18th and the 19th century where from it finite dimensional space, metric spaces, topographical spaces, and the infinite Euclidean space were formed. These conceptual spaces theoretically define limits and continuity as concerning open sets, but in their application in the real life situation, they constitute the “close to” criterion (Adams, 1999).  Majority of the content in this paper strictly focuses on the Rn space.

An example showing the continuity concept.

Lets start by investigating how a function behaves as it approaches a specific point.

Letting f (x) = sin(x).

The first thing to note is that we use radians not degrees to operate trigonometric functions i.e to obtain the derivative of sin x being cos x we use radians.

Let f (x ) = log( x ). Let f (x) when x =a, and assume f (a)= a.  Let f (x) = x if f(x) = 1 .For x Let f( x) = sin x

The above function behaves in a normal manner in the cases it is defined bearing in mind we are restricting ourselves to a specific domain. Lets investigate the limits graphically,

From the above graph we want to estimate the lim x→-2 f(x). If the estimation was to be in numeric form, the table would definitely show the happenings of the vertical coordinate with the respect of x coordinate approaching negative 2 from both sides (Landau, 2000). Luckily, use of graphs eliminates the complexity and we use the following steps:

When x<0, the graph is a parabola that faces upwards.

When x>=0, the graph is a line

We note that as x gets approaches zero the parabola tends to curve down and goes towards the origin (0, 0). But we notice when x become zero this function jumps up to (0,2) since it switches over to the line function.  

The part circle of the parabola that is open at a point near the origin, shows that the function ends when (x = 0) is not definite there either this is because this part is only defined when x<0. This clearly depicts that this function is not that smooth at the points near the origin. The smoothness aspect of the function is what is referred to as the continuity. A function can therefore be continuous or non-continuous (discontinuous) (Pickover, 2003). 

It is therefore clear that one cannot understand continuity if he or she does not understand the limits concept. A limit can be defined as the value a function attains in a random manner as one of its independent variable approaches a certain number.  Taking note of the above graph, and its explanation, I was using limits approach to find out whether the function is continuous at a certain point.
Generally limits are expressed as follows:

Lim x->c f(x) = n

This means that, as the value of x approaches value c, the total value of the function approaches n or gets to n. it is wise to note that the concern does not lie on the value of function at point c, but we are purely concerned with the value of our function as it gets close to c.  In case we are concerned with the value at c we just use the substitution method to solve this.

But since we are concerned with continuity, we have to investigate the values close to c to ensure we capture all numbers.  For example, let’s investigate a function: f(x) = x. This is a line that passes through the origin at an angle of 45 degrees. By investigating the limit as x approaches zero, we’ve our function value tending to zero also. Let investigate the function below:

This is also a line

when x=0 our function jumps to 5. But when we investigate its limit as x tends to zero, the limit is zero. This shows that that the theory of continuity is not concerned with the value of the function at zero but its value as x nears zero. These kinds of limits are called the one sided limits since they do not consider the direction of origin of the x (Tom, 1967).
The following is the two sided limits:

A) left hand limit

it x is approaching c from the left side  

Lim x->c- f(x) = n   

B) Right hand limit

X is approaching c from the right side

Lim x->c+ f(x) = n   

Continuity can therefore be termed as a property of functions that determines the smoothness of a function. In simple words, a continuous function is a function that can be plotted on the graph without any curve breaks (America., 1988).  We can now conclude that the function is continuous at every point of it if it is continuous at every single point in its domain. It is also continuous at one point if the one-sided limit at that point is equals the value of the function at that point.

Generally:

If:    lim x->c   f(x) = f(c)

Then the function can be termed as continuous at point [c, f(c)]. In order to consider the function’s continuity, we must therefore evaluate the limits from both sides. If the two limits happen not to be equal then the one-sided limit does not exist. Rewriting a better definition,

If:  lim x->c-   f(x) = lim x->c+   f(x) = f(c).  Then the function is said to be continuous at point [c, f(c). In the case of polynomials, they are continuous by meaning. Rational functions are said to be continuous in all their domains unless when given that the denominator is zero since this will give indeterminate results. This also holds true for the sum, difference, product, and quotient of two or more continuous functions. Trigonometric functions are continuous in all points except possibly at the undefined domain (Thomas/Finney, 1996). Taking the following as an example,

We want to check continuity at x=0, 

Evaluating both the two sided limits:

Lim   f(x) = 02 + 3 = 3

x->0.  Lim   f(x) = -02 + 3 = 3

x->0+In this example the limits from both sides are equal which implies that the  function might be continuous, but we have to investigate if the graph is continuous at the point x=0 before making our conclusions. Thus, after checking if both the two-sided limits are equal we have to check if they equal the value of the function at the given point:

Lim x->0 f(x) = 3

Lim X->0 f(x) = f (0)?

We use f(x) which is defined at point x=0, we evaluate it with (-x2 + 3) to evaluate f(0):  F(0) = -0^2 + 3 = 3. This function therefore satisfies all conditions of continuity. It has the similar two-sided limit at the point and the limits equal the value of the function at the point. In conclusion, the following example clearly outlines the continuity: A function f(x) is said to be continuous at a point x=a of its domain, if and only if f(x)= f(a).Thus f(x) is continuous at x=a f(x)= f(a)    f(x) = f(a). If f(x) is not continuous at a point x=a, then it is said to be discontinuous at x=a.  Given the graph of f(x), decide if f(x) is continuous at, , and .

Solution: We seek to get the limits at the point and the value of that function at the very point. If they happen to be equal then the function is continuous at that point and if otherwise, the function is discontinuous at that point. For       .

Lets investigate the epsilon definition of limits:

For every e >= 0, there exists & >= 0 such that |f(x) −f(x0)| < =e whenever |x −x0| < =&. E depends on &, but also on c. So if e is particularly small, we might need a smaller &, but if x0 were a certain number, we also might need a smaller &. But sometimes, for every e, you can find & that works, no matter what is the value of x0 is. This is what we call uniform continuity.