Differentiability, and why it forces continuity while the reverse fails

What to assume they know

• The derivative at a point as a limit of a difference quotient, from Class XI
• The derivative as a function, and the standard notations for it
• The sum, product and quotient rules from Class XI
• Continuity at a point, and the three demands hidden in Definition 1
• One-sided limits, and what it means for one of them not to be finite
• The modulus and greatest integer functions, and their two-branch and step forms
• The algebra of limits, in particular the limit of a product

What they should be able to do

• State the definition of the derivative at a point and say what the attached caution is guarding against
• Restate differentiability at a point as two one-sided limits that are finite and equal, and treat those as two separate checks
• Say what differentiability on a closed interval demands, and what is used at each end
• Reproduce Theorem 3's proof and identify the single algebraic step it turns on
• Quote Corollary 1 and produce a function refuting the reversed implication
• Compute the two one-sided difference quotients of the modulus at zero and conclude non-differentiability
• Show that a step function fails to be differentiable at a named integer, distinguishing a limit that disagrees from a limit that is not finite
• Construct a function continuous everywhere and non-differentiable at exactly two named inputs
• Read the four standard derivatives in Table 5.3 and say which class of function each covers

Where it usually goes wrong

• "Continuous means differentiable." Corollary 1 runs one way. The chapter refutes the reverse on the same page with the modulus, and Exercise 5.2 Q9 asks the student to do it again.
• "Differentiable means continuous, so I can skip checking continuity." That is the correct use of the theorem and it is worth saying out loud, because it saves work: a function known differentiable at an input needs no separate continuity check there.
• "Not differentiable means the two slopes disagree." That is one of the two ways. Exercise 5.2 Q10 fails because one of the two quotients is not finite at all. A student who only knows the first failure mode will write down two numbers that do not exist.
• "The derivative at a point is the slope of the graph, so a graph with no break has a derivative." The chapter's definition is a limit, not a picture. The modulus has no break and no derivative at zero.
• "On a closed interval you need both one-sided derivatives at the ends." Only the one that lies inside the interval. The other would be asking about inputs outside the domain.
• "Theorem 3 needs the function to be differentiable near the point." It needs it at the point. The proof consumes the hypothesis once, and only to know that one limit is a real number.
• "A function that fails at infinitely many inputs is a different sort of object." The staircase fails at every integer by exactly the argument used at one input. Nothing new is needed.
• "Any function built from moduli is non-differentiable everywhere." Two shifted moduli added together fail at exactly two inputs and are perfectly differentiable at every other. Miscellaneous Exercise Q20 turns on that.

Questions to check understanding

• State the two-part condition for differentiability at a point and apply it to a named function
• Prove that a shifted modulus is not differentiable at the shift — the form of Exercise 5.2 Q9
• Show a step rule non-differentiable at a named integer, and say which half of the two-part condition fails — the form of Exercise 5.2 Q10
• Reproduce the proof of Theorem 3 and name the step where the hypothesis is used
• Decide, for a function stated to be differentiable at a point, whether it is continuous there, and justify the answer in one line
• Construct a function continuous everywhere and non-differentiable at exactly two inputs, and justify both halves — the form of Miscellaneous Exercise Q20
• Quote the four entries of Table 5.3 and use the power entry on a polynomial

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 recall block opening §5.3 (Part I pp. 118–119). It restates the derivative at a point as a limit of a difference quotient with a small increment, gives the two notations for it, extends the same limit to a function of the input, lists the three notations for the derivative function, and names the process. The phrase to keep is the caution attached to the definition: the limit is said to exist and the whole of §5.3 is what happens when it does not.
• The three algebra-of-derivatives rules (Part I p. 119). Sum and difference; the product rule, which the chapter also names after Leibnitz; and the quotient rule with its proviso that the divisor not vanish. All three are quoted from the previous class and none is proved here.
• Table 5.3 (Part I p. 119). Four standard derivatives: a general whole power, the sine, the cosine and the tangent. Verified: the four entries are the usual ones and the power rule is stated with a general exponent, so it alone covers every polynomial. This is the chapter's only numbered table after Part I p. 109, and the whole of §5.3 to §5.7 is built on it plus two more derivatives that arrive in §5.4.
• The definition of differentiability at a point (Part I p. 119). The chapter spells it out as two one-sided difference-quotient limits that are finite and equal. Verified as a two-part test by the chapter's own later use: the modulus at zero fails the equality half with both halves finite, while the step function of Exercise 5.2 Q10 fails the finiteness half outright. Two different failure modes, one definition, and a student who reads it as a single condition will describe the second case wrongly.
• Differentiability on an interval (Part I p. 119). Every point of a closed interval, with the two ends handled by the single side that lies inside. A printed slip a reviewer should see is in this sentence — see Notes. The content is not in doubt: at the lower end only the quotient from above is available, at the upper end only the one from below.
• Theorem 3 and its proof (Part I p. 120). The statement: differentiability at an input implies continuity there. The proof rewrites the change in output as the difference quotient multiplied by the change in input, takes limits, and splits the limit of the product into a product of limits. Verified: the second factor tends to zero and the first tends to the derivative, which is a real number by hypothesis, so the product tends to zero and the output's limit equals its value. The rewrite in the second line is the whole proof; the hypothesis is used exactly once, to know the first factor is finite.
• Corollary 1 (Part I p. 120). The one-line version, quantified over the domain: differentiability anywhere on it brings continuity with it. This is the form most exercises quote, and the Summary on Part I p. 146 repeats it with the warning about the reversed implication attached.
• The converse, refuted (Part I p. 120). The modulus is continuous, by the earlier Example 7. The chapter computes the difference quotient at zero from below and from above. Verified: below zero the modulus of a small negative increment is the increment negated, so the quotient is minus one; above zero it is the increment itself, so the quotient is one. Both are finite; they differ; the derivative at zero therefore does not exist. Note the shape of the refutation — it does not show the function is badly behaved, only that one equality fails at one input.
• Exercise 5.2 Q9 (Part I p. 122). Prove that a shifted modulus fails at the shift. Verified: the same two quotients as the chapter's own case, translated along the line — minus one from below and one from above, both finite and unequal. This is the exercise where the student first reproduces the argument themselves.
• Exercise 5.2 Q10 (Part I p. 122). The staircase rule, restricted to the open interval from zero to three, is to be shown non-differentiable at the inputs one and two. Verified, and the working is worth the time: at the input one the function takes the value one; for a small negative increment the function drops to zero, so the quotient is minus one divided by the increment, which grows past every bound as the increment shrinks — it is not finite; for a small positive increment the function is still one, so the quotient is zero. The second one-sided limit exists and the first does not, so the two-part test fails on the first part. The same happens at the input two. This is the second failure mode and the chapter never contrasts it with the first.
• Miscellaneous Exercise Q20 (Part I p. 145). Does a function exist that is continuous everywhere and fails to be differentiable at exactly two inputs? Verified: yes. Add two shifted moduli with different shifts. Each summand is continuous everywhere, so the sum is, by the first part of Theorem 1; away from the two shifts the sum is linear on each stretch and therefore differentiable; and at each shift the two one-sided quotients differ by two, exactly as in the chapter's own case. Exactly two is the part to check, and it is checked by observing that the sum is a straight line on each of the three stretches between and beyond the shifts. This is the natural closing section of the explanation.

Figures to have open

• A four-row table for section 5 built from Table 5.3 (Part I p. 119). Use the repo's DataTable component.
• A graph of the modulus function with two short tangent-like segments drawn at the origin, one of each slope, for sections 9 and 10. The chapter prints no figure at all in §5.3 — verified on the page image of every page from Part I p. 118 to Part I p. 125 — so this is an added drawing.
• A step graph restricted to the open interval from zero to three for section 11, with filled left ends and hollow right ends, matching the convention of Fig 5.8 on Part I p. 112.
• A graph of the sum of two shifted moduli for section 12, with the three linear stretches distinguished and the two failing inputs marked. Not in the book.
• An arrow diagram for section 8: one arrow from differentiable to continuous, and the reverse arrow drawn and struck through.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 5, §5.3 Differentiability, the recall block, Part I pp. 118–119
• The three algebra-of-derivatives rules and Table 5.3, Part I p. 119; the two-part condition and the interval definition, Part I p. 119
• Theorem 3 with its proof, Corollary 1 and the modulus refutation, Part I p. 120
• Exercise 5.2, questions 9 and 10, Part I p. 122
• Miscellaneous Exercise on Chapter 5, question 20, Part I p. 145; Summary, the differentiability bullet, Part I p. 146