Physics and Mathematics
Continuity at End Points
Concept Overview
A function is defined on an interval. If the function has endpoints (like ([a, b])), we cannot check continuity in the same way as interior points because:
- At the left endpoint (a), there is no left neighborhood.
- At the right endpoint (b), there is no right neighborhood.
So, the definition of continuity is modified at the endpoints.
Continuity at a Left Endpoint (a)
A function (f(x)) is continuous at the left endpoint (a) of an interval ([a, b]) if:
[
\lim_{x \to a^+} f(x) = f(a)
]
This means:
We check only the right-hand limit (RHL) at (x=a).
Continuity at a Right Endpoint (b)
A function (f(x)) is continuous at the right endpoint (b) of ([a, b]) if:
[
\lim_{x \to b^-} f(x) = f(b)
]
Here we check only the left-hand limit (LHL) at (x=b).
Simple Step-by-Step Examples
Example 1
Check continuity at [(x = 2)] for:
[f(x) = 3x – 1 \text{ on }] [2, 7]
Step 1: Evaluate [f(2)]
[
f(2) = 3(2) – 1 = 5
]
Step 2: Check right-hand limit
[
\lim_{x \to 2^+} (3x – 1) = 5
]
📌 Since RHL (=) value at point ⇒ Continuous at left endpoint (x=2) ✔️
Example 2
Check continuity at (x = 5) for:
[f(x)=\left\{ \begin{align}
& \begin{array}{*{35}{l}}
{{x}^{2}}-1, & x<5 & {} \\
\end{array} \\
& 4x-5,\text{ }x\ge 5 \\
\end{align} \right.]
Step 1: Value at point
[
f(5)=4(5)-5=15
]
Step 2: LHL
[\lim_{x \to 5^-} (x^2 – 1)] [=25 – 1] [=24]
LHL ≠(f(5)) ⇒ Discontinuous at right endpoint (x=5)
Key Takeaway
| Position | Continuity Check |
|---|---|
| Left endpoint (a) | Only RHL at [x\to a^+] |
| Right endpoint (b) | Only LHL at [x\to b^-] |
Conceptual Questions with Solutions
1. Why do we not consider the left-hand limit at left endpoint?
Because points less than the left endpoint are outside the domain, so the left-hand neighborhood does not exist.
2. At the right endpoint, why only LHL matters?
Because there is no value of the function to the right of the right endpoint.
3. Can a function be continuous at endpoints but discontinuous inside?
Yes. Interior points follow the full continuity condition; endpoints use one-sided limits.
4. If a function is continuous on open interval (a,b), is it continuous at endpoints?
No. Endpoint continuity must be separately checked.
5. What type of limit is used at x=a?
Right-hand limit only.
6. What type of limit is used at x=b?
Left-hand limit only.
7. Is a function defined at endpoints necessary for continuity?
Yes, otherwise continuity cannot be established.
8. Are endpoints inside the domain?
Yes, they belong to the domain when the interval is closed.
9. Does continuity at endpoints guarantee closed interval continuity?
Yes, along with continuity inside the interval.
10. Is right-hand continuity at left endpoint same as full continuity?
No, it is a special case using only one-sided limit.
11. Do jumps matter at endpoints?
Yes, if one-sided limit doesn’t match the value, a jump discontinuity occurs.
12. Can a function fail continuity at only one endpoint?
Yes, and in that case it’s not continuous on the entire closed interval.
13. Can we test continuity outside the domain?
No, continuity is considered only where the function exists.
14. If a function is continuous in (a,b), what must be checked to say continuous on [a,b]?
Continuity must hold at both endpoints using one-sided limits.
15. Is a polynomial continuous at endpoints of any interval?
Yes. Polynomials are continuous everywhere in â„ť.
FAQ / Common Misconceptions
1. “A function continuous everywhere must be continuous at endpoints automatically.”
No. Endpoints need separate one-sided limit checks.
2. “We check both LHL and RHL at endpoints.”
Incorrect. Only one-sided limit exists.
3. “Infinity can be considered part of the interval.”
No. Endpoints must be real numbers.
4. “Continuity does not require the function value to exist at endpoints.”
It **must** exist.
5. “If a limit exists from both sides, it’s continuous.”
Value at endpoint must also match that limit.
6. “Continuity definition is same everywhere.”
Modified at endpoints due to lack of neighborhood on one side.
7. “Piecewise functions are never continuous at endpoints.”
They can be, but conditions must be satisfied.
8. “Jump discontinuity cannot occur at endpoints.”
It **can** if the one-sided limit does not match the value.
9. “Endpoints do not matter in exams.”
Many questions check this exact concept!
10. “Closed interval continuity only depends on interior points.”
Wrong — endpoints are crucial.
Practice Questions
Q1
Check continuity at (x = 0) for
[f(x) = \sqrt{4 – x^2}] [\text{ on }] [-2, 0]
LHL does not exist ⇒ Only RHL
[
\lim_{x\to 0^+}\sqrt{4 – x^2} = 2
]
[
f(0) = 2
]
âś” Continuous
Q2
Check continuity at [(x = 3)] for [f(x)=x^2 – 3x + 2] on ([3,7])
RHL at ≤ interior, so at endpoint:
[\lim_{x\to 3^+} (3)=2,] ; [f(3)=2] [\Rightarrow \text{Continuous}]
Q3
[f(x) =Â {\begin{array}{*{20}{l}} \begin{array}{l} 2x + 1,\\ 4, \end{array}&\begin{array}{l} x < 1\\ x \ge 1 \end{array}&{} \end{array}}]
[\text{ on } [1,4]]
At (1):
[
\lim_{x\to 1^-}(2x+1)=3,; f(1)=4 \Rightarrow \text{Not continuous}
]
Q4
Check continuity at [(x=2)] for [f(x)=|x-2|] on ([2,5])
RHL:
[\lim_{x\to2^+}|x-2|] [=0=f(2)] [\Rightarrow \text{Continuous}]
Q5
Check continuity of [f(x)=\sin x] at endpoint of ([0, \pi/2])
[\lim_{x\to0^+}\sin x] [=0=f(0)]
âś” Continuous
Q6
Check continuity at (x=1):
[f(x)=\frac{1}{x}] on ([1,4])
[
\lim_{x\to1^+}\frac{1}{x}=1=f(1)
]
âś” Continuous
Q7
[f(x) =Â {\begin{array}{*{20}{l}} \begin{array}{l} x + 3,\\ 7, \end{array}&\begin{array}{l} x < 4\\ x \ge 4 \end{array}&{} \end{array}}]
[\text{ on } [4,10]]
At (4):
[\lim_{x\to4^-}(4+3)] [=7=f(4)]
âś” Continuous
Q8
Check continuity of [f(x)=\ln(x+1)] at (-1) on ([-1,3])
[\lim_{x\to -1^+}\ln(x+1)] [=\ln(0)=-\infty]
Not equal to finite [f(-1)] (undefined) ⇒ ❌ Discontinuous
Q9
Check continuity at endpoint [x=5] of ([2,5]):
[
f(x)=(x-5)^2
]
[\lim_{x\to5^-}(0)] [=0=f(5)] [\Rightarrow \text{Continuous}]
Q10
$ \displaystyle f(x)=\left\{ \begin{array}{l}x+3,\text{ }x<4\\7,\text{ }x\ge 4\end{array} \right.$
[\text{ on } [-3,0]]
At (0):
[
\lim_{x\to0^-}(x^2)=0=f(0)
\Rightarrow \text{Continuous}
]
âś” Continuity at endpoints uses one-sided limits
âś” Endpoints matter a lot in exam questions
âś” Always compare limit = function value
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