Unit 2: Differentiation — Definition & Fundamental Properties

Differentiability & Continuity

Relating differentiability to continuity and identifying where a function fails to be differentiable.

Advanced25 min lesson4 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

The Difference Quotient

Every Differentiable Function Is Continuous — Not the Reverse

A tangent line assumes there's a single, well-defined direction the graph is heading at that point. Before reading on: could a function be continuous — no jumps, no holes — and still fail to have a tangent line at some point?

If f'(a) exists, something guaranteed follows: f must be continuous at a. Here's why. If limₕ→0 [f(a+h) − f(a)]/h = f'(a), a finite number, then f(a+h) − f(a) can be rewritten as h · [f(a+h) − f(a)]/h — and as h → 0, this is (something going to 0) times (something going to the finite number f'(a)), which goes to 0. So limₕ→0 f(a+h) = f(a), which is exactly the statement that f is continuous at a. Differentiability is the stronger condition; continuity is a necessary consequence, not an independent assumption.

The reverse implication fails, though: a function can be continuous at a point with no tangent line existing there at all. This happens at a corner (the one-sided derivatives exist but disagree), a cusp (both one-sided slopes go to infinity, but with opposite sign), or a vertical tangent (both one-sided slopes go to infinity with the same sign, so technically no finite derivative exists).

Worked Example — A Corner: Continuous but Not Differentiable

f(x) = |x| is continuous at x = 0 (f(0) = 0, and limₓ→0 |x| = 0 — they match). Check differentiability using one-sided derivatives: from the left, limₕ→0⁻ [|h| − 0]/h = limₕ→0⁻ (−h)/h = −1 (since h < 0 means |h| = −h). From the right, limₕ→0⁺ [|h| − 0]/h = limₕ→0⁺ h/h = 1. The one-sided derivatives disagree (−1 ≠ 1), so f'(0) does not exist — a corner, despite f being perfectly continuous there.

Worked Example — A Cusp: A Different Kind of Sharp Point

f(x) = x^(2/3) is continuous at x = 0 (it equals (x^(1/3))², always ≥ 0, and equals 0 at x = 0). Its derivative, f'(x) = (2/3)x^(−1/3), is undefined at x = 0. Checking the one-sided behavior: as x → 0⁺, x^(−1/3) → +∞, so f'(x) → +∞; as x → 0⁻, x^(1/3) is negative, making x^(−1/3) negative and large in magnitude, so f'(x) → −∞. Both one-sided slopes blow up, but with opposite signs — a cusp, a sharper failure of differentiability than a corner's finite but disagreeing slopes.

Derivative Explorer

f(x) =

x^3 - 3*x

Point of tangency

x = a1

Show tangent line

Trace mode

f(x), with tangent line at x = a

Show f''(x)Show f'''(x)

Derivative graph — linked to the graph above; panning or zooming either moves both

What's happening at x = a

At x = 1, f(x) ≈ -2 — the point (1, -2).
The derivative there is f'(1) ≈ 0: the slope of the tangent line, and the instantaneous rate of change of f at this exact point.
That slope is positive, so f is increasing at this point.
The second derivative is positive here, so f is concave up (curving upward) near this point.
Near x = 1, the tangent line y ≈ -2 + 0·(x − 1) is f's best straight-line approximation — the core idea a derivative captures.

Numerical derivative comparison

Forward, backward, and central difference estimates of f'(a) as the step size h shrinks — all three should converge toward the same value.
h	Forward	Backward	Central
1	4	-2	1
0.1	0.31	-0.29	0.01
0.01	0.0301	-0.0299	0.0001
0.001	0.003	-0.003	0

Analysis of f(x)

y-intercept: (0, 0)
x-intercepts: (-1.73, 0), (0, 0), (1.73, 0)
Extrema: local max at (-1, 2); local min at (1, -2)
Inflection points: (0, 0)
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-970, 970]

Analysis of f'(x) — its roots are f's critical points

y-intercept: (0, -3)
x-intercepts: (-1, 0), (1, 0)
Extrema: local min at (0, -3)
Inflection points: none found in view
Vertical asymptotes: none found in view
Horizontal asymptotes: none found
Domain: all real numbers in view
Range (estimated): approximately [-3, 297]

Tip

Continuity is necessary for differentiability, but never sufficient — before assuming a continuous function's derivative exists everywhere, check for corners, cusps, and vertical tangents, especially at any point defined piecewise or involving an absolute value or a fractional power.

Common Mistakes

Assuming that because a function is continuous everywhere, it must also be differentiable everywhere.
Continuity only rules out jumps, holes, and asymptotes — it says nothing about corners, cusps, or vertical tangents, all of which are perfectly continuous but not differentiable.
Concluding a function is discontinuous at a point just because it isn't differentiable there.
Non-differentiability doesn't imply discontinuity — |x| is a clean example of a function that's continuous everywhere but fails to be differentiable at exactly one point.

Key Takeaways

Differentiability at a point guarantees continuity there, proven directly from the difference quotient's limit.
Continuity does not guarantee differentiability — a function can be continuous at a corner, a cusp, or a vertical tangent and still have no derivative there.
One-sided derivatives that disagree (finite values) signal a corner; one-sided derivatives that both diverge to infinity signal a cusp or vertical tangent, depending on whether the signs agree.

Summary

Knowing when a derivative exists sets up the next lessons, which develop fast, reliable rules for actually computing derivatives — starting with the power, sum, and difference rules.

Track your progress

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