MathematicsChapter 2: Limit, Continuity and Derivative
Continuity At A Point
f is continuous at a when f(a) exists, lim(x->a)f(x) exists, and both are equal.
Learning objectives
- State the rule for continuity at a point.
- Solve a continuity at a point problem step by step.
- Check the result, restrictions, notation, and units.
Definition
f is continuous at a when f(a) exists, lim(x->a)f(x) exists, and both are equal.
Formula
k=4 in f(x)
Where / symbols
- Continuity at a point
- f is continuous at a when f(a) exists, lim(x->a)f(x) exists, and both are equal.
CONCEPT AND RULE
f is continuous at a when f(a) exists, lim(x->a)f(x) exists, and both are equal.
METHOD
For continuity at a point, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.
WORKED EXAMPLE
Choose k=4 in f(x)=x2-4x-2 for x!=2 and f(2)=k.
COMMON MISTAKE
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
How to solve continuity at a point
- Identify the mathematical objects, givens, restrictions, and required result.
- Choose and write the relevant definition, theorem, or formula.
- Substitute with brackets and carry out one justified step per line.
- Simplify, state the final answer clearly, and verify it in the original setting.
Worked example
- Worked example:
- Choose k=4 in f(x)=x2-4x-2 for x!=2 and f(2)=k.
In simple words
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Key points
- Continuity at a point
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Continuity At A Point
Sticky Notes · Formula and Worked Revision
Core Idea
- f is continuous at a when f(a) exists, lim(x->a)f(x) exists, and both are equal.
Rule / Formula
- f is continuous at a when f(a) exists, lim(x->a)f(x) exists, and both are equal.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- Choose k=4 in f(x)=x2-4x-2 for x!=2 and f(2)=k.
Common Error
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
FBISE Exam Tip
- Show the formula, substitutions, algebra, restrictions, and final answer; method marks need visible reasoning.
60-second revision
- f is continuous at a when f(a) exists, lim(x->a)f(x) exists, and both are equal.
- Worked pattern: Choose k=4 in f(x)=x2-4x-2 for x!=2 and f(2)=k.
- Final check: domain, sign, notation, units, and reasonableness.
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