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Ask an AI tutor whether a function is continuous or differentiable at a point and you get a confident paragraph in seconds. What matters is whether it did the checks. Here is how to make it show its working, and two functions that catch it skipping a step.
What should an AI tutor check for continuity and differentiability at a point?
NCERT's Class 12 Mathematics textbook, Chapter 5 (Continuity and Differentiability), states the test plainly. A function f is continuous at x = c if the left hand limit, the right hand limit and the value f(c) all exist and are equal.
Differentiability needs one more layer. The derivative at c is the limit of [f(c + h) − f(c)] / h as h → 0, so the left hand derivative (h → 0 from below) and the right hand derivative (h → 0 from above) must both exist and agree.
The same chapter proves that a function differentiable at c is continuous at c, and shows that the converse is false, with f(x) = |x| as the counterexample (NCERT, Theorem 3 and Corollary 1).
So a complete answer has five items: left limit, right limit, f(c), left derivative, right derivative. If one is missing, the AI skipped a check.
Why test the AI instead of just reading its answer?
A field experiment published in PNAS in 2025 gave nearly a thousand high school maths students access to GPT-4 tutors (Bastani et al., PNAS). During practice, grades improved by 48% with a plain ChatGPT-style interface and by 127% with a version whose prompts were designed to safeguard learning. When access was taken away, students who had used the plain version scored 17% lower than students who never had it; the safeguards largely mitigated that harm.
So make the AI lay out each check, and test it on cases where you already know the answer; that keeps you doing the reasoning. Our post on rating your confidence before checking an AI answer builds the same habit.
What prompt should you give the AI?
Copy this, then fill in the function and the point:
"Check whether f is continuous at x = c. Find the left hand limit, the right hand limit and f(c) separately, and show each calculation."
"Now find the left hand and right hand derivatives at c using [f(c + h) − f(c)] / h, with h → 0 from each side. Use the actual value f(c), not a limit."
"State both conclusions and tell me which of the five numbers decided each one."
"Do not give the final verdict until I have written my own guess."
Worked test 1: the corner f(x) = |x| at x = 0
Write down your own answers first, then compare.
Left hand limit: −x → 0. Right hand limit: x → 0. f(0) = 0. All equal, so |x| is continuous at 0.
Left hand derivative: (|h| − 0)/h = −h/h = −1. Right hand derivative: h/h = 1.
They disagree, so |x| is not differentiable at 0, exactly as NCERT works it out.
Watch for "|x| is not differentiable, so it is not continuous", which runs the theorem backwards.
Worked test 2: the jump at x = 0
Take f(x) = 1 for x ≤ 0 and f(x) = 2 for x > 0, the first example in the same NCERT chapter.
Left hand limit 1, right hand limit 2. They differ, so f is not continuous at 0.
Left hand derivative: (1 − 1)/h = 0.
Right hand derivative: (2 − 1)/h = 1/h, which grows without bound as h → 0 from above, so it does not exist.
This is the trap. Both pieces are flat, so a careless answer says "slope 0 on both sides, so the derivative is 0", using each piece's slope instead of the difference quotient with f(0) = 1. Since differentiable implies continuous, the failed continuity check already settles it. If the AI misses this, push back: "Use f(0) in the right hand quotient. What happens?"
How do you know the AI got it right?
Run this checklist after every answer:
Did it compute all five quantities separately?
Did it use f(c), not a limit, inside the derivative quotient?
Did it apply "differentiable implies continuous" in the right direction?
Did its verdicts on |x| and the jump match yours before you trusted it on a new function?
Picturing slopes makes this easier; sketching the derivative from a function graph and the derivative as the slope of a tangent both help.
Fermi's AI tutor teaches rather than answers: it decides whether to explain, ask or show, draws the figure and checks understanding as it goes, so the left and right checks happen with you, not for you. For more practice, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning.
If the AI passes both tests, try it on f(x) = |x − 1| at x = 1 from NCERT's Exercise 5.2.
