Continuity of Piecewise Functions: LHL, RHL, and f(a)

Piecewise functions can look more frightening than they really are.

The long bracket, the different rules, and the conditions like x < 2, x = 2, and x > 2 make many students feel that every line has to be checked at once. That is where mistakes begin.

Continuity questions become much easier when you treat them like a three-part matching test.

At the point where the rule changes, ask:

1. What is the left-hand limit?
2. What is the right-hand limit?
3. What is the actual value of the function at that point?

If all three match, the function is continuous at that point. If even one does not match, the function is not continuous there.

This guide will show you the exact order to follow, the branch to use in each case, and the common traps that cause wrong answers.

What Continuity Means at One Point

A function is continuous at x = a when there is no break, jump, hole, or sudden mismatch at that point.

In classroom language, the graph should pass through the point smoothly enough that the value near a and the value at a agree.

There are three conditions:

Check	Meaning
f(a) is defined	The function has an actual value at x = a
lim f(x) as x -> a exists	The left side and right side approach the same value
lim f(x) = f(a)	The approaching value matches the actual function value

For piecewise functions, the second condition is the most important one to handle carefully.

Why? Because the rule may be different on the left and right side of the same point.

What LHL and RHL Mean

LHL means left-hand limit.

It asks:

As x approaches a from values smaller than a, what value does f(x) approach?

So for LHL, use the rule that applies just before a.

RHL means right-hand limit.

It asks:

As x approaches a from values greater than a, what value does f(x) approach?

So for RHL, use the rule that applies just after a.

The actual value f(a) is different. It asks:

When x is exactly equal to a, what is the function value?

That means you must use the branch that includes the equality sign at x = a.

This one distinction prevents most errors in continuity questions.

The Right Order to Check Continuity

Use this order every time:

1. Find the point where the definition changes.
2. Find f(a) using the branch that contains x = a.
3. Find LHL using the branch just to the left of a.
4. Find RHL using the branch just to the right of a.
5. Compare all three values.

You may wonder why f(a) is checked before the two limits. It is not compulsory, but it is practical. It forces you to notice which branch actually defines the value at the point. That is often where the equality sign matters.

A Clean Answer Format

For a function tested at x = a, write your answer like this:

At x = a:

f(a) = ...

LHL = lim f(x) as x -> a from the left = ...

RHL = lim f(x) as x -> a from the right = ...

Since LHL = RHL = f(a), f is continuous at x = a.

Or, if they do not match:

Since LHL, RHL, and f(a) are not all equal, f is not continuous at x = a.

This format is simple, but it is powerful. It shows the examiner exactly what you checked.

Example 1: A Piecewise Function That Is Continuous

Consider:

f(x) = x + 1,  if x < 2
     = 3,      if x = 2
     = 5 - x,  if x > 2

Test continuity at x = 2.

First find the actual function value.

Since the middle branch says x = 2:

f(2) = 3

Now find the left-hand limit.

For values smaller than 2, use:

f(x) = x + 1

So:

LHL = 2 + 1 = 3

Now find the right-hand limit.

For values greater than 2, use:

f(x) = 5 - x

So:

RHL = 5 - 2 = 3

Now compare:

LHL = 3
RHL = 3
f(2) = 3

All three are equal.

Therefore, the function is continuous at x = 2.

Example 2: When LHL and RHL Do Not Match

Consider:

f(x) = x^2,      if x < 1
     = 2x + 1,   if x >= 1

Test continuity at x = 1.

First find f(1).

Since x >= 1 includes x = 1, use:

f(x) = 2x + 1

So:

f(1) = 2(1) + 1 = 3

Now find LHL.

For values smaller than 1, use:

f(x) = x^2

So:

LHL = 1^2 = 1

Now find RHL.

For values greater than 1, use:

f(x) = 2x + 1

So:

RHL = 2(1) + 1 = 3

Compare:

LHL = 1
RHL = 3
f(1) = 3

Here, LHL and RHL are not equal.

Therefore, the limit as x -> 1 does not exist, and the function is not continuous at x = 1.

Notice something important: f(1) and RHL are equal, but that is not enough. LHL must also match.

Example 3: When the Limit Exists but f(a) Is Different

Consider:

f(x) = x + 2,  if x < 3
     = 10,     if x = 3
     = 8 - x,  if x > 3

Test continuity at x = 3.

First find f(3).

The middle branch gives:

f(3) = 10

Now find LHL.

For values smaller than 3:

f(x) = x + 2

So:

LHL = 3 + 2 = 5

Now find RHL.

For values greater than 3:

f(x) = 8 - x

So:

RHL = 8 - 3 = 5

Compare:

LHL = 5
RHL = 5
f(3) = 10

The left-hand and right-hand limits are equal, so the limit exists.

But the actual function value is different.

Therefore, the function is not continuous at x = 3.

This is a common type of error in answers. Students see that LHL and RHL match and immediately write “continuous”. But one more check is still left.

Example 4: Finding a Constant for Continuity

Many questions ask you to find a missing constant so that the function becomes continuous.

Consider:

f(x) = kx + 2,  if x < 3
     = 11,      if x = 3
     = x^2 + 2, if x > 3

Find k so that f(x) is continuous at x = 3.

For continuity, we need:

LHL = RHL = f(3)

First find f(3):

f(3) = 11

Now find RHL:

RHL = 3^2 + 2 = 11

Now find LHL:

LHL = k(3) + 2 = 3k + 2

For continuity:

3k + 2 = 11

So:

3k = 9
k = 3

Therefore, the value of k is 3.

Where Piecewise Continuity Usually Needs Checking

You do not need to test every possible value of x unless the question specifically asks you to discuss continuity over a larger interval.

Most standard pieces like polynomials are already continuous within their own intervals.

So the real work is usually at the joining point, where the definition changes.

For example:

f(x) = x + 4,   if x < 5
     = 2x - 1,  if x >= 5

The definition changes at x = 5.

So the main continuity check is at x = 5.

Similarly:

f(x) = x^2 + 1, if x <= -2
     = 3x + 7,  if x > -2

The definition changes at x = -2.

So check continuity at x = -2.

How to Read the Inequality Signs

The inequality signs tell you which rule to use.

Condition	Use it for
x < a	LHL only
x <= a	LHL and f(a)
x = a	f(a) only
x >= a	RHL and f(a)
x > a	RHL only

This table is worth remembering. It is often the difference between a correct answer and a confused one.

Here is the main idea:

• Strict inequalities like < and > help with approaching from one side.
• Inequalities with = can give the actual value at the point.
• A separate x = a branch gives only f(a), not LHL or RHL.

Common Mistakes in Piecewise Continuity

Mistake 1: Using the middle value for the limit

If a function says:

f(x) = 7, if x = 2

that value is only f(2).

It does not decide what happens as x approaches 2 from the left or right.

For LHL and RHL, use the branches on either side.

Mistake 2: Ignoring the equality sign

If a branch says:

x <= 4

then that branch includes x = 4.

So it is used for f(4) and for the left-hand limit.

If another branch says:

x > 4

then it is used for the right-hand limit only.

Mistake 3: Stopping after LHL equals RHL

LHL and RHL matching only proves that the limit exists.

Continuity needs one more match:

common limit = f(a)

If the function value is placed somewhere else, the graph has a hole-like mismatch at that point.

Mistake 4: Saying “not defined” too quickly

Sometimes students do not see a separate x = a line and assume f(a) is not defined.

Check the inequality signs first.

If one branch says x <= a or x >= a, then f(a) is defined by that branch.

Mistake 5: Checking the wrong point

Continuity questions often hide the joining point in the conditions.

Look at the number beside the inequality sign. That is usually the point to check.

For example:

x < -1
x >= -1

The joining point is x = -1, not x = 1.

A Fast Continuity Checklist

Before writing the final conclusion, run through this checklist:

Question	If yes
Did I identify the joining point correctly?	Continue
Did I find f(a) using the equality branch?	Continue
Did I use the left branch for LHL?	Continue
Did I use the right branch for RHL?	Continue
Are LHL, RHL, and f(a) all equal?	The function is continuous at x = a

If the answer to the last question is no, the function is not continuous at that point.

How to Write the Final Line

Your final conclusion should be direct.

If the function is continuous, write:

Since LHL = RHL = f(a), the function is continuous at x = a.

If LHL and RHL are different, write:

Since LHL is not equal to RHL, the limit does not exist. Hence the function is not continuous at x = a.

If LHL and RHL are equal but f(a) is different, write:

Since the common limit is not equal to f(a), the function is not continuous at x = a.

These lines are short, but they show the logic clearly.

Practice Set

Try these mentally first, then write the full LHL, RHL, and f(a) comparison.

Question 1

f(x) = 2x + 1, if x < 2
     = 5,      if x = 2
     = x + 3,  if x > 2

At x = 2, the function is continuous because:

LHL = 5, RHL = 5, f(2) = 5

Question 2

f(x) = x + 1,  if x < 0
     = 4,      if x = 0
     = 1 - x,  if x > 0

At x = 0, the function is not continuous because:

LHL = 1, RHL = 1, f(0) = 4

The limit exists, but it does not match the actual function value.

Question 3

f(x) = ax - 1, if x < 4
     = 7,      if x = 4
     = x + 3,  if x > 4

Find a for continuity at x = 4.

For continuity:

RHL = 4 + 3 = 7
f(4) = 7
LHL = 4a - 1

So:

4a - 1 = 7
4a = 8
a = 2

The Simple Way to Remember It

Think of the joining point as a meeting place.

The left side arrives there. The right side arrives there. The actual function value is waiting there.

If all three meet at the same height, the function is continuous.

If one of them is somewhere else, there is a break in continuity.

So whenever you see a piecewise function, do not panic. Find the joining point, read the equality sign, calculate LHL and RHL calmly, and compare all three values.

That is the whole method.

Frequently Asked Questions

What is the condition for continuity of a piecewise function at x = a?

The condition is:

LHL = RHL = f(a)

This means the left-hand limit, right-hand limit, and actual value of the function at a must all be equal.

Is LHL the same as f(a)?

No. LHL is the value approached from the left side of a. f(a) is the actual value when x is exactly equal to a. They may be equal, but they are not the same idea.

Which branch should I use for f(a)?

Use the branch that includes equality at a. Look for x = a, x <= a, or x >= a.

Which branch should I use for LHL and RHL?

For LHL, use the rule that applies to values just less than a. For RHL, use the rule that applies to values just greater than a.

If LHL and RHL are equal, is the function always continuous?

No. LHL and RHL being equal means the limit exists. The function is continuous only if that common limit also equals f(a).

What if f(a) is not defined?

Then the function is not continuous at x = a. A function must have an actual value at the point to be continuous there.

Do I need to check continuity at every value of x?

Usually, no. In a piecewise function made of simple expressions, focus first on the point where the rule changes. That is where continuity is most likely to fail.

What is the most common mistake in these questions?

The most common mistake is using the x = a branch to find LHL or RHL. The x = a branch gives only the actual value f(a). The limits come from the left and right branches.