Understanding Continuity and Discontinuities

The Three Pillars of Continuity

The intuitive idea of an unbroken graph is a great starting point. In physics, for example, we naturally assume that the position of a particle changes continuously over time; it doesn't just vanish from one point and reappear at another. Calculus provides a precise way to express this idea. For a function $f(x)$ to be continuous at a point $x=a$, three specific conditions must be met.

Let's build these conditions from the ground up by examining why a function might not be continuous.

As you've just read, the three conditions can be summarized as:

1. $f(a)$ is defined. (There must be a point on the graph at $x=a$.)
2. ${\lim }_{x\to a}f(x)$ exists. (The graph must approach the same height from both the left and the right.)
3. ${\lim }_{x\to a}f(x)=f(a)$. (The height the graph is approaching must be the actual height of the point.)

When all three conditions hold, the function is continuous at $x=a$. This entire set of conditions is elegantly captured in a single, powerful statement:

A function $f(x)$ is continuous at a point $a$ if ${\lim }_{x\to a}f(x)=f(a)$.

A function that fails to meet these conditions at a point $a$ is said to have a discontinuity at that point.

Classifying the Breaks: Types of Discontinuities

For the JEE, it's not enough to know if a function is discontinuous; you must also classify the type of discontinuity. The classification follows a clear, logical flowchart based on which of the continuity conditions fail.

The first and most important question to ask is: Does the limit exist at the point of discontinuity?

• If ${\lim }_{x\to a}f(x)$ exists (and is a finite number), the discontinuity is Removable.
• If ${\lim }_{x\to a}f(x)$ does not exist, the discontinuity is Non-Removable.

The following video from Unacademy's JEE channel provides a fantastic, exam-oriented overview of this classification system. It presents the logic as a clear decision tree, which should resonate with your coding experience.

Let's formalize and summarize the types of discontinuities you just saw.

1. Removable Discontinuity

This occurs when ${\lim }_{x\to a}f(x)$ exists, but is not equal to $f(a)$. This can happen in two ways:

• Missing Point: $f(a)$ is undefined. There's a "hole" in the graph that could be "plugged."
  • Example: $f(x)=\frac{x^2-4}{x-2}$ at $x=2$. The limit is 4, but $f(2)$ is undefined.
• Isolated Point: $f(a)$ is defined, but has a different value from the limit. The point exists but is "out of place."
  • Example: $f(x)=\{\begin{array}{ll}x+2,& x\ne 1\\ 5,& x=1\end{array}$ at $x=1$. The limit is 3, but $f(1)=5$.

2. Non-Removable Discontinuity

This occurs when ${\lim }_{x\to a}f(x)$ does not exist. There are three primary reasons for this.

• Jump Discontinuity: The left-hand and right-hand limits both exist and are finite, but they are not equal (${\lim }_{x\to a^{-}}f(x)\ne {\lim }_{x\to a^{+}}f(x)$). The function "jumps" from one value to another. The magnitude of this jump, $|\text{LHL}-\text{RHL}|$, is called the jump of the discontinuity.
  • Example: The greatest integer function, $f(x)=[x]$, at any integer value.
• Infinite Discontinuity: At least one of the one-sided limits tends to $+\infty$ or $-\infty$. This is visually represented by a vertical asymptote.
  • Example: $f(x)=\frac{1}{x-2}$ at $x=2$.
• Oscillating Discontinuity: The function values oscillate so rapidly near the point that they don't approach any single number.
  • Example: $f(x)=\sin (\frac{1}{x})$ at $x=0$.

The following resource provides clear, formal definitions and worked-out examples for each of these categories.

Continuity Over an Interval

We've focused on continuity at a single point, but we often need to discuss continuity over a whole range of values, or an interval.

Open Interval: A function is continuous on an open interval $(a,b)$ if it is continuous at every single point within that interval. This is a straightforward extension of our point-wise definition.

Closed Interval: Continuity on a closed interval $[a,b]$ is a bit more subtle because we can't take a two-sided limit at the endpoints $a$ and $b$. Here, we use one-sided limits. A function is continuous on $[a,b]$ if:

1. It is continuous on the open interval $(a,b)$.
2. It is continuous from the right at $a$. This means ${\lim }_{x\to a^{+}}f(x)=f(a)$.
3. It is continuous from the left at $b$. This means ${\lim }_{x\to b^{-}}f(x)=f(b)$.

The following Khan Academy video gives an excellent intuitive and graphical explanation of this concept.

Conclusion

In this lesson, we have formally defined continuity and established a systematic way to analyze and classify discontinuities. This brings our foundational module on limits to a close. You are now equipped with a robust toolkit for analyzing the behavior of functions near a point.

Key Takeaways:

• A function $f$ is continuous at a point $a$ if the limit exists, the function is defined at the point, and the limit equals the function's value: ${\lim }_{x\to a}f(x)=f(a)$.
• Discontinuities are classified by checking if the limit exists. If it does, the discontinuity is removable (missing or isolated point). If it doesn't, it is non-removable (jump, infinite, or oscillating).
• Continuity on a closed interval $[a,b]$ requires continuity on the interior $(a,b)$ plus continuity from the right at $a$ and from the left at $b$.

In our next lesson, we will explore one of the most important consequences of continuity: the Intermediate Value Theorem. This theorem allows us to make powerful statements about the values a function must take on, and it provides a method for proving the existence of solutions to equations—a valuable tool in both mathematics and physics.