Undergraduate → Real Analysis → Functions of Real Variables ↓
Limits and Continuity
Introduction
In real analysis, the concepts of limit and continuity are fundamental. They provide the basis for calculus and allow us to understand how functions behave near points of interest. The idea of limit involves determining the value that a function approaches as it approaches a value of the input. Continuity is based on limits, defining when a function can be drawn without lifting the pen from the paper.
Understanding the limitations
The limit is the value that the function f(x) approaches as x approaches a number. It is a way of discussing what happens as we get closer to a certain point, without necessarily saying that we will reach that point.
Formal definition of limit
Let's define what it means mathematically for a function to have a limit. Suppose f(x) is a function, and a is a point within its domain. We say that the limit of f(x) as x approaches a is L, and we write:
lim (x → a) f(x) = l
This means that for every number ε > 0, no matter how small, there exists a number δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε
Example 1: Limit of a constant function
Consider the constant function f(x) = 5 We can find the limit as x approaches any point a.
lim (x → a) f(x) = 5
The value of the function is always 5, regardless of x, so the limit is 5.
One-sided limits
Sometimes it is useful to look at the limit from only one side, either the left or the right. These are called one-sided limits.
- The right-hand limit is written as:
lim (x → a + ) f(x) - The left limit is written as:
lim (x → a - ) f(x)
Example 2: One-sided limit
Consider a function defined piecewise:
f(x) = {
2x + 1, if x < 2;
3x – 1, if x ≥ 2.
,
To find the one-sided limit at x = 2 :
- lim (x → 2 - ) f(x) = 2(2) + 1 = 5
- lim (x → 2 + ) f(x) = 3(2) - 1 = 5
Both one-sided limits are equal to 5, so the two-sided limit at x = 2 is also 5.
Visualizing the limits
Using graphs can help to understand limits intuitively. Consider the graph of the function as you approach a point a.
In this diagram, as x approaches a from either side, the value of the function approaches the height of the red dot. This height represents the limit.
Continuation of works
Continuity means smoothness. Intuitively, a continuous function means you can graph it without lifting your pen. Mathematically, a function f(x) is continuous at a point a if:
f(a)is defined.lim (x → a) f(x)exists.lim (x → a) f(x) = f(a).
Example 3: Polynomial
Polynomials such as f(x) = x^2 + 2x + 1 are continuous everywhere in their domain. For any real number a, f(a) is defined, and the limit as x → a is simply the evaluation of the polynomial at a.
Discontinuities
If a function fails to be continuous at a point, it is discontinuous. There are several types of discontinuity:
- Point discontinuity: The limit exists, but
f(a)is either not defined or is not equal to the limit. - Jump discontinuity: the left and right limits exist, but are not equal.
- Infinite discontinuity: The function approaches infinity as
xapproachesa.
Example 4: Step function
Consider the step function:
f(x) = {
1, if x < 0;
2, if x ≥ 0.
,
At x = 0, the limit does not exist because:
- lim (x → 0 - ) f(x) = 1
- lim (x → 0 + ) f(x) = 2
Thus, f(x) has a jump discontinuity at x = 0.
The idea of continuity
Continuity can be seen by connecting every point of the graph smoothly. Below is a visualization:
The graph is continuous at a (blue point) as there is no break or hole, while there is a break at b (red point), indicating discontinuity.
Conclusion
Limits and continuity are important concepts in real analysis. Limits help us understand the behavior of a function near specific points, while continuity ensures that functions behave in a predictable manner without any abrupt changes. These foundations pave the way for advanced topics like derivatives and integrals in calculus. Understanding their nuances and how they are interrelated is the key to mastering the field of real analysis.
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