A function is continuous where its limit and its value agree

What to assume they know

• Limits from Class XI: what a function approaches near an input, and the two one-sided versions of that question
• The algebra of limits — sums, products and quotients of limits
• Piecewise definitions, and reading which branch applies at a given input
• Domain, range, and what a function returns at a given input
• Polynomial, constant, identity and reciprocal functions and their graphs
• Interval notation, and the difference between a closed and an open end
• The idea that a symbol can be undefined at a point without the formula around it being wrong

What they should be able to do

• Split Definition 1 into its three separate demands and say which one a given function fails
• Compute both one-sided limits of a piecewise function at the join and compare them with the value there
• Decide continuity at a stated point for a function given by a single formula, by evaluating the limit and the value
• Distinguish a mismatch of the two one-sided limits from a mismatch between the limit and the value
• State Definition 2 and explain why it is quantified over the domain rather than over the real line
• Apply the endpoint form of continuity on a closed interval, using only the side that exists
• Explain why a function whose domain is a single point is continuous
• Read the two tables on Part I p. 109 and say why the reciprocal function's break at zero cannot be repaired by assigning a value
• Explain why an infinite one-sided limit means the limit does not exist
• Solve for an unknown constant that makes a piecewise function continuous at a stated point, and recognise the case where no constant works

Where it usually goes wrong

• "Continuous means you can draw it without lifting the pen." The chapter offers that picture and immediately calls it naive, then replaces it with Definition 1. Two pages later, on Part I p. 111, it calls a function continuous whose graph cannot be drawn in one stroke. The picture is a memory aid for one common case, not the definition.
• "If the limit exists, the function is continuous." Existence of the limit is one demand of three. Example 4 has a perfectly good limit at zero and is still not continuous there, because the value assigned at zero is a different number.
• "If the value exists, the function is continuous." The first opening function is defined at zero and still fails, because the two sides disagree. Being defined is the cheapest of the three demands.
• "A hole in the domain is a discontinuity." Continuity is only ever asserted at points of the domain. The reciprocal function is called continuous on Part I p. 108 and it has no value at zero at all, so zero is not a point where it can fail.
• "The limit at zero is infinity, so the limit is infinity." The chapter writes the symbol and then denies, in the next clause, that any real number has been named, concluding that the one-sided limit fails to exist. An explanation that leaves the notation in view without the retraction teaches the error.
• "Both one-sided limits agreeing settles it." They settle demand two. The value still has to match, which is exactly what the second opening function fails.
• "At the end of a closed interval you compare both sides." Only one side is available. The chapter says the other is not meaningful there, and a student who looks for it will invent a limit outside the domain.
• "There must be some constant that patches any join." Exercise 5.1 Q18 has none, because the branch carrying the unknown vanishes at the join whatever the unknown is. Reaching for a formula rather than comparing the two sides is what hides this.

Questions to check understanding

• State the three conditions in Definition 1 and name which one a given function fails at a stated input
• Compute the two one-sided limits of a two-branch function at its join and decide continuity there
• Check continuity of a single-formula function at a named input — the form of Exercise 5.1 Q1, Q2 and Q4
• Classify a function that is undefined at a point as continuous on its domain rather than discontinuous at that point — the form of Exercise 5.1 Q3(b) and Q3(c)
• Find the constant that makes a two-branch function continuous at a named input — the form of Exercise 5.1 Q27 to Q29
• Find two constants making a three-branch function continuous — the form of Exercise 5.1 Q30
• Recognise a join at which no constant can produce continuity, and justify the claim — the form of Exercise 5.1 Q18
• Explain why an infinite one-sided limit is a statement that the limit fails to exist

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its exercises.

• The first opening function and Fig 5.1 (§5.2, Part I pp. 104–105). Two constant branches, one on the inputs at most zero and one on the inputs above zero. Read off the figure at three hundred dots per inch: the upper branch begins at a hollow circle on the vertical axis and runs right; the lower branch runs in from the left and ends at a filled dot on the axis. Neither branch carries an arrowhead. Verified: approaching zero from the left gives the lower constant, from the right the upper, and the value at zero is the lower one — so demand two of the three fails and demand three would have been satisfied had it been reached. The chapter's own remark is that the value there matches the left side.
• The second opening function and Fig 5.2 (§5.2, Part I p. 105). One constant everywhere except at zero, where a larger value is assigned. Read off a three-hundred-dot the printed page: a horizontal line with arrowheads at both ends, carrying a hollow circle where it crosses the vertical axis, and a filled dot directly above it. Verified: both one-sided limits exist and agree, so demand two passes; demand three fails, and mending it would take a single reassignment. This pair is the whole of section 1 and the two figures must be shown together, because separately each looks like the same accident.
• Definition 1 (§5.2, Part I p. 105). The chapter names the point, requires it to lie in the domain, and sets the limit equal to the value. It then restates the condition in the longer form that names the two sides and the value as three things that must exist and coincide. Use the longer form; the short one is the compression, not the meaning.
• Examples 1, 2 and 3 (Part I p. 106). A linear rule at input one, a squaring rule at zero, and a modulus at zero. Verified: the linear rule takes the value five at input one and its limit there is five; the squaring rule takes zero at zero and its limit is zero; the modulus is split into two branches and both one-sided limits are zero, as is the value. All three are continuous, and the third is the only one where the two sides have to be computed separately.
• Example 4 (Part I pp. 106–107). A cubic plus three away from zero, and the value one at zero. Verified: the limit at zero is three and the value is one, so the function is not continuous there, and zero is its only bad input. This is the cleanest instance in the chapter of demand three failing on its own.
• Examples 5 and 6 (Part I p. 107). A constant function and the identity function, each shown continuous at every real number in about four lines. Verified: both arguments are one substitution long. Their value is that they are the base cases everything in the next topic is assembled from.
• Definition 2 and its elaboration (Part I p. 107). The definition quantifies continuity at a point over every point of the domain. The elaboration then handles a closed interval: at the left end only the limit from the right is compared with the value, at the right end only the limit from the left, and the chapter says outright that the two missing one-sided limits are not meaningful there. The final sentence is the one to show: a function whose domain is one point is continuous, because the single demand is satisfied vacuously.
• Examples 8 and 9 (Part I p. 108). A cubic-plus-square-minus-one, unbroken at every input whatever; and the reciprocal function, unbroken at every point of its domain — which excludes zero. Verified: the reciprocal's limit at any non-zero input is the reciprocal of that input, which is its value there. The chapter calls the reciprocal a continuous function. Say that sentence slowly: the function is continuous and its graph is in two pieces.
• Tables 5.1 and 5.2 with Fig 5.3 (Part I p. 109). Two tables of the reciprocal's values, the first at inputs shrinking to zero from above and the second from below, both ending in a column written as a power of ten. Verified: the tabulated values are exactly the reciprocals of the tabulated inputs, and the two columns marked with a general exponent are consistent with the six explicit ones. The figure is the two-branch hyperbola with eight points labelled — four on each branch — joined to the axes by dotted guides.
• The infinity paragraph (Part I p. 109). The chapter writes each one-sided limit with an infinity symbol and then says, in capitals, that no real number is being named — from which it draws the conclusion that neither one-sided limit exists at all. This is a trap worth thirty seconds: the notation looks like an answer and the sentence after it withdraws the answer.
• Exercise 5.1 Q1 to Q5 (Part I p. 116). Q1 checks a linear rule at three named inputs. Q2 checks a quadratic at one input. Q3 has four parts: a linear rule; a reciprocal shifted by five; a difference of squares over a linear factor; and a shifted modulus. Q4 puts a power function at the input equal to its own exponent. Q5 is a two-branch function checked at three inputs. Verified: Q1 and Q2 are continuous at every input named, as is Q4 since a power function is a polynomial. In Q3, (a) and (d) are unbroken across the whole line, while (b) and (c) are continuous at every point of the domains stated and are simply not defined at the excluded input — that distinction is section 7 of the explanation, and Q3 is where a student first meets it. In Q5 the function is continuous at zero and at two, and not at one, where the two sides give one and five.
• Exercise 5.1 Q17, Q18 and Q30 (Part I pp. 117–118). Q17 asks for a relation between two constants making a two-branch function continuous at three. Q18 asks for a constant making a two-branch function continuous at zero, and then asks about the input one. Q30 asks for two constants making a three-branch function continuous everywhere. Verified: Q17 forces three times the first constant plus one to equal three times the second plus three, so the first exceeds the second by two thirds. Q18 is the interesting one — the left branch has limit zero at zero and the right branch has limit one, and the left branch carries the unknown as a factor of a quantity that vanishes, so no value of the constant works; at the input one the function is continuous whatever the constant is, because only the right branch is in play there. Q30 gives two linear conditions in two unknowns, solved by the pair two and one. Q18 belongs in the explanation: an exercise whose honest answer is "no such value".
• Exercise 5.1 Q26 to Q29 (Part I p. 118). Four items each asking for the one constant that makes a two-branch function continuous at a named input. Verified: Q27 needs four times the constant to equal three, giving three quarters; Q28 needs the constant times pi plus one to equal minus one, giving minus two over pi; Q29 needs five times the constant plus one to equal ten, giving nine fifths. Q26 is different and the difference is the point: it needs the limit of a cosine over a linear expression vanishing at the same place, which reduces after one substitution to a standard limit — the ratio of a sine to its own argument — that this chapter never states. Class XI supplies it. The answer is six. Flag the dependency in the script; do not attribute the limit to this chapter.

Figures to have open

• Redraws of Fig 5.1 and Fig 5.2 (Part I pp. 104–105) with the marker fills exactly as listed above. These two carry sections 1 and 2 and the whole point is that they differ: the first has the hollow marker above and the filled one below, the second has the filled marker above and the hollow one on the line.
• A single shown step by step number line for section 3, with one pointer approaching from each side and a readout of the arriving value on each side.
• A redraw of Fig 5.3 (Part I p. 109): the two-branch reciprocal curve with the eight labelled points and their dotted guides. The chapter's own figure.
• A two-row table for section 11 built from Tables 5.1 and 5.2 (Part I p. 109). The content is the chapter's; the layout is added here. Use the repo's DataTable component.
• A closed-interval diagram for section 8 showing the two ends with only the inward side live. Not in the book; the chapter draws nothing here.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 5 "Continuity and Differentiability", §5.1 Introduction, Part I p. 104
• §5.2 Continuity, the two opening functions with Fig 5.1 and Fig 5.2, Part I pp. 104–105; Definition 1, Part I p. 105
• Examples 1 to 4, Part I pp. 106–107; Examples 5 and 6 with Definition 2 and its elaboration, Part I p. 107
• Examples 7 to 9, Part I p. 108; Tables 5.1 and 5.2 with Fig 5.3 and the infinity paragraph, Part I p. 109
• Exercise 5.1, questions 1 to 5, Part I p. 116; questions 17, 18 and 20, Part I p. 117; questions 26 to 30, Part I p. 118