Continuity at a Point

A function is continuous at a point if there is no sudden jump, break, or hole 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.

A function f(x) is said to be continuous at a point x = c if the following three conditions are satisfied:

1. f(c) is defined: The function must have a value at x = c.
2. The left-hand limit and right-hand limit must both exist and be equal, i.e., \lim\limits_{x \to c^+} f(x) = \lim\limits_{x \to c^-} f(x)
3. The limit of the function as x approaches a must equal the actual value of the function at a, i.e.,\lim\limits_{x \to c} f(x) = f(c) .

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

Example: 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:

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.

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.

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.

Situations Where Continuity Fails

A function f(x) is not continuous at x=a if any one of the following conditions fails:

1. Function is not defined at the point:The function has no value at x=a(creates a hole or gap) .i.e., f(a) does not exist
2. Limit does not exist: The left-hand and right-hand limits are not equal (jump discontinuity) .i.e., \lim_{x \to a^-} f(x) \ne \lim_{x \to a^+} f(x)
3. Limit is not equal to function value: The limit exists, but it does not match the function value (removable discontinuity).i.e., \lim_{x \to a} f(x) \ne f(a)
4. Function becomes unbounded (infinite discontinuity): The function approaches infinity near the point (vertical asymptote) .i.e., \lim_{x \to a} f(x) = \infty \ \text{or} \ -\infty

Solved Examples

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

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.

Check if g(1) exists:
g(1) = \frac{1^2 - 1}{1 - 1} = \frac{0}{0}, which is indeterminate and hence 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?

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.

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.

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