To show that $e^x$ is continuous, we need to verify that the function satisfies the mathematical definition of continuity. A function $f(x)$ is continuous at a point $c$ if:

$\lim_{x \to c} f(x) = f(c)$

1. Define the Function

The exponential function $f(x) = e^x$ is defined as:

$e^x = \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n$

This definition ensures that $e^x$ is well-defined for all real numbers $x$.

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2. Check the Limit

To show continuity at any point $c$, evaluate:

$\lim_{x \to c} e^x = e^c$

Proof Outline:

• Step 1: Behavior Around $c$: The exponential function is smooth and differentiable for all real $x$. Hence, its value changes smoothly near $c$.

• Step 2: Substitution: Substitute the definition of $e^x$ as a limit into the continuity equation:

  $\lim_{x \to c} e^x = \lim_{x \to c} \lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n$ Swapping the limits is valid because the convergence is uniform in $x$. This simplifies to: $\lim_{n \to \infty} \left( 1 + \frac{c}{n} \right)^n = e^c$

Thus, $\lim_{x \to c} e^x = e^c$, showing $e^x$ is continuous.

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3. Differentiability Implies Continuity

The function $e^x$ is differentiable (its derivative is $e^x$), and differentiable functions are always continuous. Hence, $e^x$ is continuous for all real $x$.

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Conclusion

Since $\lim_{x \to c} e^x = e^c$ for any $c$, and the function is differentiable, $e^x$ is continuous everywhere on $\mathbb{R}$.

Would you like to see a graphical or formal epsilon-delta proof of this?

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5 Related Questions:

1. How do we prove that $e^x$ is differentiable?
2. What does the epsilon-delta definition of continuity imply for $e^x$?
3. How does $e^x$ behave as $x \to \infty$ or $x \to -\infty$?
4. Can we generalize the continuity of $e^x$ to $a^x$ for $a > 0$?
5. How do we formally define the exponential function using power series?

Tip: The exponential function $e^x$ is unique because its derivative is equal to itself, making it a central