Continuous functions
Continuity is not a technical detail: it is what lets you guarantee that a root exists, that a maximum exists and that the voltage across a capacitor cannot jump.
01Three conditions, not one
A function is continuous at a point a when all three of these hold at once:
The intuitive picture —“it can be drawn without lifting the pencil”— is fine for getting started, but it is imprecise: there are continuous functions that are impossible to draw. The definition above is the one that is used.
02The three ways to fail
The limit exists, but f(a) does not exist or has some other value. It is fixed by redefining the function at that single point.
Both one-sided limits exist and are finite, but differ. The difference is the size of the jump. It cannot be fixed in any way.
One of the one-sided limits is infinite or does not exist, as with 1/x at zero or with sin(1/x). It is the worst one.
The function is the same on both sides; what changes is what happens at the point and how it is approached.
03Which functions are continuous
The sum, difference, product and quotient of continuous functions are continuous —the quotient, where the denominator does not vanish—, and the composition of continuous functions is continuous. That is enough for almost everything, because the elementary functions are continuous on their domains:
| Function | Continuous on | Where it fails |
|---|---|---|
| Polynomial | All of R | Nowhere |
| Rational | Its domain | Where the denominator vanishes |
| Exponential and sine/cosine | All of R | Nowhere |
| Logarithm | (0, ∞) | Not defined for x ≤ 0 |
| Tangent | Its domain | At π/2 + kπ: infinite discontinuity |
| Piecewise-defined | Inside each piece | The joining points have to be studied |
That is why, in practice, studying the continuity of a function given by a formula comes down to looking at the “suspicious” points: zero denominators, changes of branch and boundaries of the domain.
04The three theorems that give power
A function that is continuous on a closed interval [a, b] has properties that are used all the time, often without being named:
If f is continuous on [a, b] and f(a) and f(b) have opposite signs, then there is at least one c between a and b with f(c) = 0. In other words: to go from negative to positive without jumping, you have to cross zero.
The generalization: f takes all the values between f(a) and f(b). If a thermometer read 18 °C in the morning and 27 °C in the afternoon, at some moment it read exactly 22.5 °C.
A function that is continuous on a closed and bounded interval attains an absolute maximum and an absolute minimum. It does not merely get close: it reaches them at some point. It is the guarantee that an optimization problem has a solution before you go out looking for it.
All three hypotheses matter. On (0, 1], which is not closed, f(x) = 1/x is continuous and has no maximum.
05Bolzano in action: the bisection method
Bolzano does not just assure you that the root exists: it tells you how to find it. If f changes sign on the interval, you split it in half and keep the half where the sign change persists. Each step halves the error.
It is slow compared with other methods, but it never fails if there is a sign change, and the error is bounded in advance: after n steps it is less than (b − a)/2n. Ten steps divide the initial interval by more than a thousand.
06Continuity in circuits
The voltage across a capacitor and the current through an inductor are continuous functions of time: they cannot jump. If the capacitor voltage jumped, the current i = C·dv/dt would be infinite; if the inductor current jumped, the voltage v = L·di/dt would be. That is the physical reason for the initial conditions used to solve any transient.
- What does jump. The voltage across a resistor, the current through a capacitor and the voltage across an inductor can change abruptly: they are the “non-continuous” variables of the circuit.
- Response speed. A real op amp cannot follow a vertical edge: its slew rate limits the slope. The output signal is still continuous, but with a ramp instead of the ideal jump.
- Schmitt trigger. Its transfer characteristic is deliberately discontinuous: it jumps between two levels, and with hysteresis, so that it does not oscillate in the presence of noise.
- Quantization. The output of an ADC is a staircase: a piecewise continuous function with jumps of one least significant bit. That jump is, precisely, the quantization error studied in A/D and D/A converters.
07In the lab
Given f(x) = (x² − 4)/(x − 2) for x ≠ 2 and f(2) = k, find the k that makes it continuous. Repeat with a piecewise function where the parameter is inside one of the branches.
Program the bisection in a spreadsheet or in C to solve the equation of the diode circuit from section 5 —a 5 V source, 1 kΩ in series and the diode with its exponential equation—, with a tolerance of 1 µA. Compare with the value given by a simulation and with the simplified 0.7 V model.
In an RC circuit, trigger a step and observe on two channels the capacitor voltage and the current. Check that the voltage does not jump and that the current does. Repeat with an inductor, swapping the roles.
08Common mistakes
- Checking only the limit and forgetting that f(a) has to exist and coincide with it.
- Calling a jump removable. If the one-sided limits differ, no redefinition can fix it.
- Applying Bolzano without a sign change, or on an interval where the function is not continuous.
- Using Weierstrass on an open interval. The hypotheses of the theorem are not decorative.
- Assuming an inductor’s current can be cut off abruptly: that is where the arc across a relay contact comes from, and why the flyback diode is fitted.
09Self-assessment
What kind of discontinuity does f(x) = (x² − 1)/(x − 1) have at x = 1?
Removable: the limit exists and equals 2; it is enough to define f(1) = 2.
And f(x) = 1/(x − 3) at x = 3?
Infinite: the one-sided limits tend to −∞ and +∞.
For what k is f(x) = 3x + k if x ≤ 2, and x² if x > 2, continuous?
Equating the one-sided limits at 2: 6 + k = 4, so k = −2.
How many bisection steps are needed to shrink an interval of length 1 to less than 0.001?
2n > 1000 → n = 10, because 210 = 1024.
Why can’t the voltage across a capacitor jump?
Because i = C·dv/dt: a jump would imply an infinite derivative and therefore infinite current, which no real source can deliver.
A signal is −2 V at t = 0 and +3 V at t = 1 ms, and the circuit is continuous. Did it pass through zero?
Yes: by Bolzano’s theorem, at some instant between the two it was exactly 0 V.
10Further reading
- James Stewart. Single Variable Calculus: Early Transcendentals. 7th ed. (Spanish edition, “Cálculo de una variable. Trascendentes tempranas”), Cengage Learning, 2012. Continuity, types of discontinuity and the intermediate value theorem with applied examples.
- Michael Spivak. Calculus. 3rd ed. (Spanish edition, “Calculus”), Reverté, 2012. Proves Bolzano, the intermediate value theorem and Weierstrass from the completeness of the reals.
- Richard L. Burden and J. Douglas Faires. Numerical Analysis. 9th ed. (Spanish edition, “Análisis numérico”), Cengage Learning, 2011. Chapter 2 compares bisection, fixed-point iteration and Newton–Raphson, with their error bounds.