# Constant Function f(x) = c: Graph, Derivative, Integral and Properties > The constant function f(x) = c: its horizontal-line graph, why its derivative is 0, its integral cx + C, limits, and common mistakes with constants. Source: https://practicecalculus.com/functions/constant-function Updated: 2026-11-03 $$ f(x) = c $$ ## At a glance - **Domain:** $(-\infty, \infty)$ - **Range:** The single value $\{c\}$ - **Derivative:** $0$ - **Integral:** $cx + C$ - **Graph:** A horizontal line through $(0, c)$ - **Slope:** $0$ everywhere ## Key takeaways - A constant function $f(x) = c$ gives the same output for every input, so its graph is a horizontal line. - The derivative of any constant is $0$, because a flat line has zero slope. - The integral of a constant is $\int c\,dx = cx + C$, the area of a rectangle of height $c$. - The lowercase $c$ in the function and the uppercase $C$ in an antiderivative are different constants. A **constant function** returns the same number no matter what you put in. Written $f(x) = c$, where $c$ is a fixed real number, it is the simplest function in calculus and the starting point for every derivative and integral rule. ## Definition and formula $$f(x) = c$$ Here $c$ can be any real number: $f(x) = 3$, $g(x) = -\tfrac{1}{2}$, $h(x) = \pi$. The input $x$ does not appear on the right-hand side, so changing $x$ never changes the output. Examples you meet in practice include a fixed fee, a constant speed's acceleration (zero), or the value $f(a)$ once $a$ is fixed. ## Graph of a constant function The graph of $y = c$ is a **horizontal line** crossing the $y$-axis at $(0, c)$. Every point on it has the same height. ```graph title: 'Graph of the constant functions y = 3 and y = -2' caption: 'Each constant function is a horizontal line. Its slope, and therefore its derivative, is $0$ everywhere.' x: [-5, 5] y: [-4, 5] functions: - expr: '3' label: y = 3 - expr: '-2' label: y = -2 points: - [0, 3, '(0, 3)'] - [0, -2, '(0, -2)'] ``` ## Derivative of a constant The derivative of a constant is zero: $$\frac{d}{dx}\, c = 0$$ From the [limit definition of the derivative](/blog/limit-definition-of-the-derivative): $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{h \to 0} \frac{c - c}{h} = \lim_{h \to 0} 0 = 0$$ Geometrically, a horizontal line has slope $0$, so every tangent line is the line itself. This fact is the [constant rule](/rules/constant-rule), and it explains why the constant term of a polynomial disappears when you differentiate. ## Integral of a constant Going backwards, a function whose derivative is $c$ is $cx$, so $$\int c\,dx = cx + C$$ For a definite integral, the area under $y = c$ from $a$ to $b$ is a rectangle with height $c$ and width $b - a$: $$\int_a^b c\,dx = c(b - a)$$ If $c$ is negative, the rectangle sits below the axis and the signed area is negative. ## Key properties - **Continuous and differentiable everywhere.** There are no breaks, holes or corners. - **Limits are trivial.** $\lim_{x \to a} c = c$ for every $a$, and also $\lim_{x \to \pm\infty} c = c$. - **Even function.** $f(-x) = f(x)$, so the graph is symmetric about the $y$-axis. - **Neither increasing nor decreasing.** It is both nonincreasing and nondecreasing. - **Every point is a maximum and a minimum.** Since the output never changes, every $x$ gives both the largest and smallest value. - **Degree zero polynomial.** A nonzero constant is $cx^0$, the bottom rung of the [polynomial functions](/functions/polynomial-functions). ## Worked examples **Example 1.** Differentiate $f(x) = 7$. The derivative is $f'(x) = 0$. **Example 2.** Differentiate $g(x) = 4x^2 + 9$. By the sum rule, $g'(x) = 8x + 0 = 8x$. The $9$ contributes nothing. **Example 3.** Differentiate $y = \pi^2$. Although it looks complicated, $\pi^2 \approx 9.87$ is just a number, so $y' = 0$. **Example 4.** Evaluate $\int_1^4 5\,dx$. The antiderivative is $5x$, so the value is $5(4) - 5(1) = 15$, the area of a $5 \times 3$ rectangle. ## Common mistakes 1. **Treating constants that look like expressions as variables.** $e^2$, $\ln 3$ and $\sin\frac{\pi}{6}$ are numbers, so their derivatives are $0$. Students often write $\frac{d}{dx}e^2 = e^2$ by mistake. 2. **Forgetting the $x$ when integrating.** $\int 5\,dx = 5x + C$, not $5 + C$. 3. **Mixing up $c$ and $C$.** The constant of integration $C$ is new and arbitrary. It is not the same number as the $c$ in $f(x) = c$. 4. **Dropping a constant factor.** The derivative of a constant *term* is $0$, but a constant *factor* stays: $\frac{d}{dx}(5x) = 5$, not $0$. See the [constant multiple rule](/rules/constant-multiple-rule). ## How it connects A constant function is a [linear function](/functions/linear-function) with slope $0$, and the special case $n = 0$ of the [power functions](/functions/power-functions). In applications, a derivative equal to zero on an interval tells you the original function is constant there, a consequence of the [mean value theorem](/theorems/mean-value-theorem). That is also why every antiderivative carries a $+\,C$: two functions with the same derivative differ by a constant. For more on this, read the guide to [antiderivatives](/blog/antiderivatives). ## Frequently asked questions ### What is the derivative of a constant? The derivative of any constant is $0$. A constant never changes, so its rate of change is zero, and the graph of $y = c$ is a flat line with slope $0$. ### What is the integral of a constant? The integral of a constant $c$ is $\int c\,dx = cx + C$. For example, $\int 5\,dx = 5x + C$, because the derivative of $5x$ is $5$. ### Is a constant function continuous? Yes. A constant function is continuous and differentiable at every real number. Its limit at any point $a$ is simply $c$. ### Is a constant function a polynomial? Yes. A nonzero constant is a polynomial of degree $0$, since $c = cx^0$. The zero function $f(x) = 0$ is also a polynomial, though its degree is usually left undefined. ### Is a constant function even or odd? Every constant function is even, because $f(-x) = c = f(x)$. Only the zero function is both even and odd.