Physics and Mathematics
Introduction to Continuity
1. Concept Overview: What is Continuity?
A function is said to be continuous at a point if its graph has no breaks, no gaps, and no sudden jumps at that point.
Imagine drawing the graph without lifting your pen — that represents continuity.
2. Mathematical Definition
A function [f(x)] is continuous at [x = a] if the following three conditions are satisfied:
1. The function value exists:
[f(a)] is defined.
2. The left-hand limit exists:
[\displaystyle \lim_{x \to a^-} f(x)]
3. The right-hand limit exists:
[\displaystyle \lim_{x \to a^+} f(x)]
And most importantly:
[\text{If } \displaystyle \lim_{x \to a^-} f(x)] [= \lim_{x \to a^+} f(x) = f(a),] [\text{ then } f(x) \text{ is continuous at } x = a.]
3. Understanding with Graph Interpretation
| Type of Situation | Graph Look | Continuity Result |
|---|---|---|
| No breaks | Smooth curve | Continuous |
| Hole in the graph | Missing point | Discontinuous |
| Jump in graph | Sudden height change | Discontinuous |
| Asymptote | Infinite break | Discontinuous |
A function can be piecewise continuous or continuous only on specific intervals.
4. Examples (Basic Understanding)
Example 1
Function: [f(x) = x + 3]
This is a polynomial function → Polynomials are continuous everywhere.
Example 2
Function: [f(x) = \dfrac{1}{x}]
Discontinuous at: [x = 0]
because denominator zero makes the function undefined.
Example 3
Piecewise function:
[f(x)] [=\begin{cases} 3x+1, & x<2 \ \text{and} \ x^2-4, & x \ge 2\end{cases}]
We must check continuity at [x = 2] using limit rule.
(Such examples will be solved in later sections.)
5. Important Theorems (Student Friendly)
1. All polynomials are continuous for all real numbers.
2. Rational functions (fractions of polynomials) are continuous
→ except where denominator = 0.
3. Trigonometric, exponential, and logarithmic functions are continuous in their domains.
This helps you quickly identify continuity without long calculations.
Conceptual Questions (With Solutions)
1. Is [f(x) = \sin x] continuous for all real x?
Yes. All trigonometric functions like sine are continuous in their domains. Since domain of sine is all real numbers, it is continuous everywhere.
2. Is [f(x) = \sqrt{x}] continuous at [-1]?
No. Because [-1] is **not in the domain** of [\sqrt{x}]. A function must exist at a point to be continuous there.
3. Can a function be continuous but not differentiable?
Yes! A sharp corner like in [f(x) = |x|] at [x = 0] is continuous but not differentiable. We will study this deeply later.
FAQ / Common Misconceptions
1. Continuous means differentiable?
No. Differentiability is stronger. Every differentiable function is continuous — but not vice-versa.
2. If left and right limits exist but aren’t equal, is it continuous?
No. They must not only exist, but **must be equal to each other and to [f(a)].**
3. What if limit exists but function value doesn’t?
Then **not continuous** — because [f(a)] must exist.
Practice Questions (With Step-by-Step Solutions)
Q1. Check the continuity of [f(x) = x^2] at [x = 3].
Solution:
- Function value: [f(3) = 9]
- LHL: [\displaystyle \lim_{x \to 3^-} x^2 = 9]
- RHL: [\displaystyle \lim_{x \to 3^+} x^2 = 9]
Since all three are equal → Continuous at [x = 3].
Q2. Check the continuity of [f(x) = \dfrac{x-1}{x-1}] at [x = 1].
Solution:
- For [x \ne 1], [f(x) = 1]
- At [x = 1], function is undefined
Although limit = 1, [f(1)] does not exist →
Not continuous at [x = 1].
Q3. Identify where [f(x) = \dfrac{x^2 – 4}{x – 2}] is discontinuous.
Solution:
- Factor numerator → [(x – 2)(x + 2)]
- For [x \ne 2], [f(x) = x + 2]
- At [x = 2], denominator = 0 → undefined
Discontinuous at [x = 2].
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