Continuity at a Point

Continuity at a point means that a function doesn’t have any sudden jumps, breaks, or holes at a particular spot. Imagine you are drawing a graph of a function on a piece of paper. If you can draw it without lifting your pencil from the paper, the function is continuous at that point.

For example, think of driving a car along a road. If the road is smooth and you don’t have to stop or make sharp turns, that’s like a function being continuous. But if there’s a gap in the road or a sudden bump, it’s like a break in the function, meaning it's not continuous at that point.

In this article, we will discuss Continuity of a function at a Point in detail.

Continuity of Function at a Point

Continuity of a function at a point refers to the behavior of the function at that point and how its value behaves in the neighbourhood of that point. A function is considered continuous at a point if it satisfies the following conditions at that point.

Conditions for Continuity at a Point

1. The function is defined at x = c.
   • This means that f(c) exists.
2. The limit of the function exists as x approaches c i.e., LHL = RHL.
   • \lim_{x \to c^+} f(x) = \lim_{x \to c^-} f(x)
3. The limit of the function as x to c must equal the actual value of the function at c, i.e., \lim_{x \to c} f(x) = f(c) .

Mathematically, this can be written as:

\lim_{x \to c} f(x) = f(c)

If any of the above conditions fail, the function is said to be discontinuous at c.

Example of Continuity

Consider the function f(x) = x². To check the continuity at x = 2:

• f(2) = 4, so the function is defined at x = 2.
• \lim_{x \to 2} f(x) = \lim_{x \to 2} x^2 = 4, so the limit exists.
• \lim_{x \to 2} f(x) = f(2) = 4, so the limit equals the function value.

Since all three conditions are satisfied, f(x) = x² is continuous at x = 2.

How to Determine Continuity at a Point?

Here are the steps to determine the continuity of a function at a point x = c:

Step 1: Check if the function is defined at x = c.

• Find the value of the function at x = c, i.e., f(c).
• If f(c) exists (i.e., it is finite), proceed to the next step.
• If f(c) does not exist, the function is not continuous at x = c.

Step 2: Check if the limit exists as x approaches c.

• Compute the left-hand \lim_{{x \to c^-}} f(x) and the right-hand limit \lim_{{x \to c^+}}.
• If both limits exist and are equal, proceed to the next step.
• If the limits do not exist or are not equal, the function is not continuous at x = c.

Step 3: Verify if the limit equals the function value at x = c.

• Check if \lim_{{x \to c}} f(x) = f(c).
• If they are equal, the function is continuous at x = c.
• If they are not equal, the function is not continuous at x = c.

If all three steps are satisfied, the function is continuous at x = c.

Solved Examples

Example 1: Determine if the function f(x) = 2x + 3 is continuous at x = 1.

Solution:

For a function to be continuous at x = 1, we need to check three conditions:

• f(1) exists.
• \lim_{{x \to 1}} f(x) exists.
• \lim_{{x \to 1}} f(x) = f(1)

Check if f(1) exists:
f(1) = 2(1) + 3 = 5.

Find \lim_{{x \to 1}} f(x):
\lim_{{x \to 1}} (2x + 3) = 2(1) + 3 = 5.

Compare the limit and the function value:
Since \lim_{{x \to 1}} f(x) = f(1) = 5, the function is continuous at x = 1.

Thus, f(x) = 2x + 3 is continuous at x = 1.

Example 2: Check if the function g(x) = \frac{x^2 - 1}{x - 1} is continuous at x = 1.

Solution:

Check if g(1) exists:
g(1) = \frac{1^2 - 1}{1 - 1} = \frac{0}{0}, which is undefined.

Find \lim_{{x \to 1}} g(x):
g(x) = \frac{(x - 1)(x + 1)}{x - 1} = x + 1 for x ≠ 1.

Now, \lim_{{x \to 1}} g(x) = \lim_{{x \to 1}} (x + 1) = 2.
Since g(1) is undefined but \lim_{{x \to 1}} g(x) = 2, the function is not continuous at x = 1.

Example 3: Is the function h(x) = |x| continuous at x = 0?

Solution:

Check if h(0) exists:
h(0) = |0| = 0.

Find \lim_{{x \to 0}} h(x):
For x > 0, h(x) = x, and for x < 0, h(x) = -x.

So, \lim_{{x \to 0^+}} h(x) = \lim_{{x \to 0^+}} x = 0, and \lim_{{x \to 0^-}} h(x) = \lim_{{x \to 0^-}} (-x) = 0.

Since both the left-hand and right-hand limits are equal, \lim_{{x \to 0}} h(x) = 0.
\lim_{{x \to 0}} h(x) = h(0) = 0, so h(x) = |x| is continuous at x = 0.

Example 4: Determine if the function f(x) = \begin{cases} x^2 & \text{if} \ x < 2 \\ 4 & \text{if} \ x = 2 \\ x + 2 & \text{if} \ x > 2 \end{cases} is continuous at x = 2.

Solution:

Check if f(2) exists: f(2) = 4.

Find \lim_{{x \to 2}} f(x):
\lim_{{x \to 2^-}} f(x) = \lim_{{x \to 2^-}} x^2 = 4, and
\lim_{{x \to 2^+}} f(x) = \lim_{{x \to 2^+}} (x + 2) = 4.

Since both the left-hand and right-hand limits are equal, \lim_{{x \to 2}} f(x) = 4.
\lim_{{x \to 2}} f(x) = f(2) = 4, so the function is continuous at x = 2.

Practice Questions

Question 1: Check whether the function f(x) = \frac{2x + 1}{x - 3} is continuous at x = 3.

Question 2: Determine if the following piecewise function is continuous at x = 1:

f(x) = \begin{cases} x^2 + 2x & \text{if} \ x < 1 \\ 3 & \text{if} \ x = 1 \\ x + 2 & \text{if} \ x > 1 \end{cases}

Question 3: Is the function g(x) = sin(x) continuous at x = π/2?

Question 4: Check the continuity of f(x) = \begin{cases} 2x - 1 & \text{if} \ x \leq 2 \\ x^2 & \text{if} \ x > 2 \end{cases} at x = 2.

Question 5: Determine if the function h(x) = \frac{x^2 - 4}{x - 2} is continuous at x = 2.

Question 6: Is the absolute value function f(x) = |x - 3| continuous at x = 3?

Question 7: Determine if the function f(x) = e^x is continuous at x = 0.

Question 8: Check the continuity of the following piecewise function at x = 0:

f(x) = \begin{cases} x^2 & \text{if} \ x < 0 \\ 0 & \text{if} \ x = 0 \\ x & \text{if} \ x > 0 \end{cases}

Answer Key

1. No, the function is not continuous at x=3.
2. Yes, the function is continuous at x=1.
3. Yes, the function is continuous at x=π/2​.
4. No, the function is not continuous at x=2.
5. No, the function is not continuous at x=2.
6. Yes, the function is continuous at x=3.
7. Yes, the function is continuous at x=0.
8. Yes, the function is continuous at x=0.

Conclusion

In simple terms, continuity at a point means that a function behaves smoothly without any breaks, jumps, or holes at that specific point.

Read More,

• Calculus in Maths
• Differential Calculus
• Limit in Calculus
• Continuity of Functions
• Differentiability of a Function
• Limits, Continuity and Differentiability