Source: Lecture notes, Analyzing Functions in Advanced Calculus
Tags: domain, range, continuity, differentiability, local behavior, global behavior, critical points, derivatives, function analysis, advanced calculus, UCF
Difficulty: Introductory-to-Intermediate Prerequisites: Comfortable with basic calculus (limits, derivatives, integrals) and function notation. If terms like "derivative" or "continuity" feel unfamiliar, review your Calculus I notes first.
This lecture lays the groundwork for how mathematicians take a function apart and understand what it does. You are moving beyond just computing derivatives and integrals and into the habit of asking structured questions about a function: where is it defined, where is it well-behaved, and where does it do something interesting? These ideas underpin nearly everything that follows in an advanced calculus sequence, from optimisation to series convergence. If you can reliably analyse domain, continuity, differentiability, and local/global behaviour, you have the toolkit for most problems in this course.
Analysing a function means systematically checking its domain and range, whether it is continuous, whether it is differentiable, and how it behaves both locally (near individual points) and globally (across its entire domain). These properties determine which calculus tools you can apply and where.
Domain
The set of all possible input values for which a function is defined. Think of it as the collection of every x-value you are allowed to plug in without breaking anything.
Range
The set of all possible output values a function can produce. In simple terms, this is every y-value the function can spit out when you feed it valid inputs.
Continuity
A function is continuous at a point if its graph has no jumps, holes, or breaks at that point. Formally, the limit of the function as x approaches that point equals the function's value there. Think of it as being able to draw the graph through that point without lifting your pen.
Differentiability
A function is differentiable at a point if it has a well-defined derivative (rate of change) at that point. The graph must be smooth there, with no sharp corners or vertical tangents. In simple terms, differentiability means the function has a clear, single slope at that point.
Derivative
A function that gives the instantaneous rate of change of the original function at each point. Notated f'(x) or df/dx. Think of it as the speedometer reading for how fast the function's output is changing.
Critical point
A value of x where f'(x) = 0 or f'(x) is undefined. These are the candidates for local maxima, local minima, or inflection points. In simple terms, it is where the function momentarily stops increasing or decreasing.
Local behaviour
How a function behaves in a small neighbourhood around a specific point. Studied using derivatives.
Global behaviour
The overall behaviour of a function across its entire domain. Studied using limits, integrals, and end-behaviour analysis.
The domain tells you which inputs are valid; the range tells you which outputs are possible.
To find the domain, look for restrictions: division by zero, square roots of negatives (in the reals), logarithms of non-positive numbers.
To find the range, consider what outputs the function can reach given its domain.
Example: f(x) = √x
Domain: x ≥ 0 (all non-negative reals), because the square root of a negative number is undefined in ℝ.
Range: f(x) ≥ 0 (all non-negative reals), because a square root always returns a non-negative value.
A function is continuous at a point if three conditions hold:
f(a) is defined
lim(x→a) f(x) exists
lim(x→a) f(x) = f(a)
A function continuous on an interval is continuous at every point in that interval.
Continuity matters because it guarantees predictable behaviour and lets you safely apply most calculus tools (derivatives, integrals, the Intermediate Value Theorem).
Example: f(x) = sin(x)
Continuous for all real numbers. The graph is smooth and unbroken everywhere.
Example: g(x) = 1/x
Not continuous at x = 0. There is a vertical asymptote there; the function blows up to ±∞ and is not defined at that point.
A function is differentiable at a point if its derivative exists there.
The derivative f'(x) gives the rate of change of f(x) at each point, which geometrically equals the slope of the tangent line.
Differentiability lets you find local extrema (maxima and minima) and understand how steeply the function is climbing or falling.
If a function is differentiable at a point, it is also continuous there. The reverse is not always true.
Example: f(x) = x²
Differentiable everywhere. Its derivative is f'(x) = 2x.
At x = 3, f'(3) = 6, meaning the function is increasing at a rate of 6 units of output per unit of input.
Local behaviour: what happens near a specific point. The primary tool is the derivative.
Global behaviour: the function's overall shape across its full domain. Tools include limits (end behaviour), integrals (accumulated change), and asymptotic analysis.
Example: f(x) = x³ – 3x² + 1
Derivative: f'(x) = 3x² – 6x
Setting f'(x) = 0: 3x(x – 2) = 0, so critical points at x = 0 and x = 2.
To classify these, check the sign of f'(x) around each critical point:
For x < 0: f'(x) > 0 (increasing)
For 0 < x < 2: f'(x) < 0 (decreasing)
For x > 2: f'(x) > 0 (increasing)
So x = 0 is a local maximum (function switches from increasing to decreasing) and x = 2 is a local minimum (switches from decreasing to increasing).
Square root function: f(x) = √x, domain [0, ∞), range [0, ∞)
Power rule derivative: if f(x) = xⁿ, then f'(x) = nxⁿ⁻¹
Derivative of x²: f'(x) = 2x
Derivative of x³ – 3x² + 1: f'(x) = 3x² – 6x
Critical point condition: set f'(x) = 0 and solve for x
Analysing domain and continuity is how engineers confirm that a model behaves sensibly across all expected inputs, for instance ensuring a control system does not hit a divide-by-zero under normal operating conditions. Finding critical points via derivatives is the basis of optimisation problems in economics (maximising profit), physics (minimising energy), and machine learning (gradient descent).
Students often assume that if a function is continuous, it must be differentiable. It need not be. The classic counterexample is f(x) = |x|, which is continuous everywhere but has a sharp corner at x = 0 where the derivative does not exist.
Confusing "not defined" with "equals zero." If f(x) = 1/x at x = 0, the function is undefined there, not equal to zero.
Thinking the range is always the same as the codomain. The range is only the set of values the function takes; the codomain is the set it is allowed to land in, and these can differ.
Assuming a critical point is always a local max or min. A critical point can also be an inflection point (e.g. f(x) = x³ at x = 0).
⚠️ Expect to be asked to find the domain and range of a function and justify your answer with reasoning about restrictions (square roots, denominators, logarithms).
⚠️ Continuity questions often ask you to identify where a function fails to be continuous and explain which of the three continuity conditions breaks down.
⚠️ Differentiability vs. continuity is a favourite exam distinction. Be ready to provide a function that is continuous but not differentiable.
⚠️ Critical point classification (local max, local min, or neither) using the first derivative test is highly likely to appear. Know the sign-change method.
True or False: Every differentiable function is also continuous.
True or False: The function f(x) = 1/x is continuous on the interval (0, ∞).
Fill in the blank: The critical points of a function occur where f'(x) = ______ or where f'(x) is ______.
True or False: The range of f(x) = √x includes negative numbers.
Fill in the blank: At a local maximum, the derivative changes sign from ______ to ______.
Answers:
True. 2. True. 3. 0; undefined. 4. False. 5. Positive; negative.
Q: What is the domain of f(x) = √(4 – x²)?
A: The expression under the square root must be non-negative: 4 – x² ≥ 0, so x² ≤ 4, giving the domain [–2, 2].
Q: Is f(x) = |x| differentiable at x = 0? Explain.
A: No. The function is continuous at x = 0 but has a sharp corner there. The left-hand derivative is –1 and the right-hand derivative is +1, so the derivative does not exist at that point.
Q: Find the critical points of f(x) = x³ – 3x² + 1 and classify each as a local maximum or minimum.
A: f'(x) = 3x² – 6x = 3x(x – 2). Setting f'(x) = 0 gives x = 0 and x = 2. Testing signs: f'(x) > 0 for x < 0, f'(x) < 0 for 0 < x < 2, f'(x) > 0 for x > 2. So x = 0 is a local maximum and x = 2 is a local minimum.
Q: Give an example of a function that is continuous everywhere but not differentiable at exactly one point.
A: f(x) = |x| is continuous for all real numbers but not differentiable at x = 0, where the graph has a corner.
Q: Why does continuity matter for applying calculus tools like the Intermediate Value Theorem?
A: The Intermediate Value Theorem requires the function to be continuous on a closed interval. If there are jumps or holes, the guarantee that the function passes through every intermediate value between f(a) and f(b) breaks down.
This material connects directly to optimisation (finding absolute max/min on closed intervals using critical points and endpoint evaluation), which appears later in most advanced calculus courses. Continuity is also a prerequisite for integration, since the Riemann integral is defined for bounded functions on closed intervals and behaves most cleanly when the function is continuous. The local vs. global behaviour distinction reappears when studying convergence of sequences and series.
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