Real Valued Functions, Limits, and Continuity
Real Valued Functions and Standard Functions
A real valued function is a function whose values are real numbers. More formally, if $A$ and $B$ are subsets of the set of real numbers ℝ, then a function $f: A → ℝ$ assigns to each element $x$ in $A$ a unique real number $f(x)$ in $ℝ$. The set $A$ is called the domain of the function, and the set of all values $f(x)$ is called the range.
Standard Functions
Standard functions are fundamental building blocks in calculus. Below is a summary of the standard functions mentioned in the syllabus:
| Function Type | General Form / Example | Domain (Typical) | Description |
|---|---|---|---|
| Polynomial Function | f(x) = anxn + an-1xn-1 + ... + a1x + a0 | (-∞, ∞) | Functions involving non-negative integer powers of $x$ and real coefficients where an ≠ 0. |
| Trigonometric Functions | sin(x), cos(x), tan(x), etc. | Varies (e.g., (-∞, ∞) for sin and cos) | Functions relating the angles of a triangle to lengths of its sides, periodic in nature. |
| Exponential Function | f(x) = ax (where a > 0, a ≠ 1) | (-∞, ∞) | Functions where the independent variable appears in the exponent. Base $e$ (Euler's number) is very common. |
| Logarithmic Function | f(x) = loga(x) (where a > 0, a ≠ 1) | (0, ∞) | The inverse of the exponential function. |
Definition: Real Valued Function
Let $A ⊆ ℝ$. A rule $f$ that associates to each $x ∈ A$ a unique real number $f(x)$ is called a real valued function of a real variable.
Exam-Oriented Notes: Always check the domain restrictions when dealing with standard functions. For instance, logarithmic functions require arguments strictly greater than zero, and rational polynomial functions require denominators to be non-zero.
Definition and Algebra of Limits
The concept of a limit describes the behavior of a function as the independent variable approaches a specific value, rather than necessarily at that exact value.
Definition: Limit of a Function
We write limx → c f(x) = L if, as $x$ gets arbitrarily close to $c$ (from both sides), the values of $f(x)$ get arbitrarily close to $L$.
Algebra of Limits (Definition Only)
If limx → c f(x) = L and limx → c g(x) = M, where $L$ and $M$ exist as real numbers, then the algebraic properties of limits are:
- Sum Rule: limx → c [f(x) + g(x)] = L + M = limx → c f(x) + limx → c g(x)
- Difference Rule: limx → c [f(x) - g(x)] = L - M = limx → c f(x) - limx → c g(x)
- Product Rule: limx → c [f(x) × g(x)] = L × M = [limx → c f(x)] × [limx → c g(x)]
- Quotient Rule: limx → c [f(x) / g(x)] = L / M, provided that M ≠ 0
- Constant Multiple Rule: limx → c [k × f(x)] = k × L, where $k$ is a constant
Related Problems on Limit: Limit evaluation problems often involve direct substitution, factoring, rationalization, or utilizing standard algebraic limit theorems. Common Mistake: Forgetting to check if the denominator approaches zero before applying the quotient rule.
Definition of Continuity
Informally, a continuous function is one whose graph can be drawn without lifting the pen from the paper. There are no sudden jumps, breaks, or holes.
Definition: Continuity of Real Valued Functions
A real valued function $f$ is continuous at a point $x = c$ in its domain if the following three conditions are met:
1. $f(c)$ is defined (c lies in the domain of $f$).
2. limx → c f(x) exists.
3. limx → c f(x) = f(c).
If a function is not continuous at a point $c$, it is said to be discontinuous at $c$.
Important Observations:
- Polynomial functions are continuous everywhere on their domain.
- Standard trigonometric, exponential, and logarithmic functions are continuous on their respective domains.
- When solving problems, always verify all three conditions of continuity sequentially.