Checking Function Continuity

1. What Does It Mean for a Function to be Continuous?

Intuitively, a function is continuous if you can draw its graph without lifting your pen from the paper. There are no sudden jumps, holes, or breaks.

This intuitive idea is formalized using the concept of limits we studied previously. For a function to be continuous at a point, three conditions must be met. The video below from Dinesh Sir gives a great, quick overview of this core principle.

The following reading provides the formal, three-part definition that you will use to solve problems.

To summarize, a function $f(x)$ is continuous at a point $x=a$ if and only if:

1. $f(a)$ is defined. (The point exists).
2. ${\lim }_{x\to a}f(x)$ exists. (This means the Left-Hand Limit and Right-Hand Limit are equal: $\text{LHL}=\text{RHL}$).
3. ${\lim }_{x\to a}f(x)=f(a)$. (The limit equals the value).

In short: LHL = RHL = Value of the function. This is the master key to solving all continuity problems.

2. Testing for Continuity at a Point

Most MHT-CET questions will give you a piecewise function and state that it is continuous. Your task will be to find the value of an unknown constant, like $k$ or $a$. The strategy is to apply the LHL = RHL = f(a) condition.

Let's see this in action. The video below explains the concept and then jumps into a typical MHT-CET problem.

As you saw, checking for continuity often boils down to solving a limit problem. Here is another example that involves a logarithmic function, requiring a slightly different approach.

3. Continuity Over an Interval

Sometimes, you'll be asked about a function's continuity over an entire interval, such as $[0,\pi ]$. This is particularly common for piecewise functions that have different definitions for different parts of the domain.

A function is continuous over an interval if it's continuous at every point in that interval. For piecewise functions, you only need to check for continuity at the break points — the points where the function's definition changes.

The following video demonstrates this perfectly. It involves finding two unknown constants, $a$ and $b$, by setting up two separate continuity equations at the two break points.

4. Special Functions: Modulus and Greatest Integer

Some functions have inherent points of interest for continuity.

• Modulus Function, $|x|$: The modulus function itself is continuous everywhere. However, when it appears in piecewise definitions or fractions, you must analyze its behavior. Remember:

  • For $x>a$, $|x-a|=x-a$.
  • For $x<a$, $|x-a|=-(x-a)$.
    This change in definition is crucial for evaluating Left-Hand and Right-Hand Limits.

• Greatest Integer Function, $[x]$: This function is discontinuous at every integer. It "jumps" up by 1 at each integer value. This is a fact you should memorize.

The video below quickly recaps these properties and solves problems involving the modulus function.

Conclusion

In this lesson, we have built upon our knowledge of limits to understand and test for continuity. You've learned the formal definition and, more importantly, the practical methods to solve the kinds of continuity problems you will face in the MHT-CET.

Key Takeaways:

• The core condition for continuity at a point $x=a$ is LHL = RHL = f(a).
• For piecewise functions given as continuous, use this condition to set up equations and solve for unknown constants.
• To check for continuity over an interval, test for continuity at the "break points" where the function's definition changes.
• Remember that the greatest integer function $[x]$ is discontinuous at all integers, and be prepared to split the modulus function $|x-a|$ into its positive and negative cases for LHL and RHL.

In our next lesson, we will take the next logical step to differentiability. We'll learn that continuity is a prerequisite for a function to be differentiable and begin exploring the rules for finding derivatives of more complex functions, such as composite and implicit functions.