DPST1013 Chap.6 Functions, Limits and Continuity
Functions, Limits and Continuity
A formula becomes a function only after its permitted inputs and intended outputs are understood. This chapter develops that starting point through inequalities, domains, limits and continuity. A square root imposes a nonnegative radicand over the real numbers, a denominator excludes its zeros, and a composition must send each allowed input into the domain of the next function.
These restrictions remain relevant after algebraic simplification. Limits describe the value approached near an input, while continuity also asks what happens at the input itself. That distinction explains why a removable hole may leave a limit intact and why assigning a different value at one point can destroy continuity without changing nearby behaviour.
One-sided limits are useful when formulas change or the domain has an endpoint. A two-sided limit requires the compatible approaches to agree. Existence theorems turn these ideas into mathematical conclusions. The intermediate value theorem uses continuity on a closed interval to guarantee an intermediate output; the maximum-minimum theorem guarantees attained bounds on that kind of interval.
Neither theorem supplies every detail of the answer by itself. An existence argument does not automatically prove uniqueness, and a guaranteed extremum still has to be located by another method. The worked question below separates those roles rather than treating a convincing plot as proof.
What this chapter covers
- 01
Real number sets, interval endpoints and the logical direction of inequalities
- 02
Domains of sums, quotients and compositions, including restrictions that survive cancellation
- 03
Finite and infinite limits, one-sided approaches and the role of a comparison bound
- 04
Continuity at a point and on an interval, including piecewise joins and removable holes
- 05
Intermediate values, attained extrema and the additional evidence needed for uniqueness
Guarantee a root before approximating it
- 2A polynomial is continuous on [0,1]. Its endpoint values are f(0)=−2 and f(1)=1, so zero lies strictly between them. The intermediate value theorem therefore gives at least one zero in the open interval (0,1).
- 2Differentiate to obtain f′(x)=3x²+2, which is positive for every real x. The mean value theorem implies that f is strictly increasing, so two different inputs cannot both give output zero. This supplies uniqueness in addition to the existence argument.
- 1At x=3/4, calculate 27/64+3/2−2=−5/64. At x=7/8, calculate 343/512+7/4−2=215/512. Their signs differ, so the zero lies between 3/4 and 7/8. The interval length is 7/8−3/4=1/8.
- 1Report an exact bracket, not a fictitious exact root: 3/4<r<7/8. Continuity justifies the bracket, and strict increase makes it the location of the only zero. Neither endpoint is itself the root because both function values are nonzero.
Key terms
- Domain
- The domain is the collection of inputs for which a function is defined. Determine it from the original expression and the stated problem context. Algebraically equivalent formulas may display different apparent domains when a factor has been cancelled, so record exclusions before simplifying.
- Range
- The range is the collection of outputs actually attained by a function on its domain. It depends on the permitted inputs as well as the formula. An upper bound approached near an excluded endpoint need not belong to the range.
- Composition
- Composition applies one function to the output of another. For f composed with g, the input must first be in the domain of g, and the resulting value g(x) must lie in the domain of f. These are two separate admissibility checks.
- One-sided limit
- A one-sided limit describes the output approached from just one direction along the real line. It is particularly useful at a piecewise join. When both sides of an interior point are available, agreement of the two one-sided limits is necessary for a two-sided limit.
- Removable discontinuity
- A removable discontinuity is a failure of continuity that can be repaired by assigning the limiting value at the affected input. The nearby formula determines the limit, while the original function value is either missing or inconsistent with it.
- Closed interval
- A closed interval includes both of its finite endpoints. That inclusion is part of the hypotheses in the intermediate value and maximum-minimum theorems used here. Removing an endpoint can remove attainment of a bound, even when the formula itself remains continuous.
Functions, Limits and Continuity FAQ
Why can a limit exist at a point outside the domain?
A limit concerns the function values at nearby permitted inputs, rather than requiring evaluation at the target input. For example, cancellation may reveal a simple expression describing every nearby value even though the original denominator is zero at the target. Keep the original exclusion when discussing the function, and use the simplified nearby expression only to establish the limit.
Does a sign change prove that a root is unique?
A sign change at the endpoints, together with continuity across the interval, proves that a root exists between them. A function may cross the axis more than once, so uniqueness needs a separate argument. Strict monotonicity is one useful route, but it must be established over the entire interval in question, not inferred from two sample values.
How should I handle an inequality with a variable denominator?
First identify where the denominator vanishes and exclude those inputs. Multiplying by a denominator of unknown sign can reverse the inequality in some regions and preserve it in others, so an unconditional multiplication is unsafe. Factor where possible, divide the line at zeros and excluded points, and determine the sign on each resulting interval.
What is the difference between a bound and an attained maximum?
A bound limits how large the outputs can be, but it does not necessarily equal any output. An attained maximum requires an allowed input whose function value is at least every other value. On a closed bounded interval, continuity guarantees that maximum and minimum values are attained. If a hypothesis fails, inspect the particular function rather than applying the theorem automatically.
When is a graph useful in a continuity question?
A graph can reveal likely jumps, holes and asymptotes, helping you choose the inputs that need closer inspection. It cannot guarantee what happens between sampled points or at an excluded endpoint. Use the graph to form a question, then verify the domain, the relevant limits and the actual function value algebraically. That sequence gives the visual evidence a precise role.
Exam move
Practise domain and range as separate tasks before combining them with limits. For each expression, list the restrictions imposed by denominators, roots and compositions, then test a permitted and an excluded value. At a piecewise join, make a small record containing the left limit, right limit and actual value; comparing those three entries is more reliable than saying that the pieces look connected.
For theorem questions, write the interval and hypotheses before the conclusion. Label a proof of existence explicitly, then ask whether the problem also requests uniqueness or an approximation. Keep exact fractions during sign checks so that rounding cannot manufacture a sign change. Finish by identifying which inputs belong to the final set and whether its endpoints are included.
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