Continuity of exponential functions (epsilon-delta)

In summary, the conversation discusses the definition of continuity and how it can be applied to the function f(x) = a^x, where a>0 is a real number. The epsilon-delta definition is suggested as a way to prove continuity, but the exact value of delta in terms of a, epsilon, and x_0 is unclear. The idea of reducing the problem to a simpler case, such as x = 0, is also mentioned.
  • #1
mattmns
1,128
6
Here is the question:
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Show the following: Let a>0 be a real number, then f:R->R defined by [itex]f(x) = a^x[/itex] is continuous.
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First we have the following definitions of continuity:
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Let X be a subset of R, let f:X->R be a function, and let [itex]x_0[/itex] be an element of X. Then the following three statements are logically equivalent:

(a) f is continuous at [itex]x_0[/itex]

(b) For every sequence [itex](a_{n})_{n=0}^{\infty}[/itex] consisting of elements of X with [itex]\lim_{n\rightarrow \infty}a_{n} = x_0 [/itex], we have [itex]\lim_{n\rightarrow \infty}f(a_n) = f(x_0)[/itex]

(c) For every [itex] \epsilon > 0[/itex] there exists a [itex]\delta > 0[/itex] such that [itex]|f(x) - f(x_0)| < \epsilon [/itex] for all x in X with [itex]|x - x_0| < \delta[/itex]

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I thought the best way to do this would be to use the epsilon-delta definition, since I really could not think of anything with the sequences (contradiction, maybe?).

So now the question is, what do we make delta. Clearly delta must depend on epsilon and on a (and I think on [itex]x_0[/itex]). What I tried was the following, but it does not look pretty:

Let [itex]x_0[/itex] be some point in R. Our function is increasing, so I said that if we look at [itex]x_0 - \delta[/itex] and [itex]x_0 + \delta[/itex] then we want to show the following: [itex]|max(a^{x_0 - \delta},a^{x_0 + \delta}) - a^{x_0}| < \epsilon[/itex] I then factored out the [itex]a^{x_0}[/itex] and started moving things around to get delta in terms of a,epsilon, and [itex]x_0[/itex], but it did not really work out, and I am not sure where to go from here.

Any ideas? Thanks!

edit... I think I remember hearing something about showing that it is continuous for some explicit point (like x = 0) and then reducing every other case (meaning every other value x can take on) to the same case of x = 0, thus showing continuity. Would this be sufficient (I think so), and would that be a good way to solve this exercise?
 
Last edited:
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  • #2
Start by showing a^x is continuous as a function from the rationals to the reals. How is a^x defined when x is irrational?
 

1. What is the definition of continuity for exponential functions?

The definition of continuity for exponential functions is that a function f(x) is continuous at a point x = a if the limit of f(x) as x approaches a exists and is equal to f(a).

2. How do you prove continuity of an exponential function using the epsilon-delta definition?

To prove continuity of an exponential function using the epsilon-delta definition, we must show that for any given epsilon, there exists a delta such that if the distance between x and a is less than delta, then the distance between f(x) and f(a) is less than epsilon.

3. Can an exponential function be both continuous and discontinuous at the same time?

No, an exponential function cannot be both continuous and discontinuous at the same time. A function can either be continuous or discontinuous at a given point, not both.

4. What are some common examples of discontinuous exponential functions?

Some common examples of discontinuous exponential functions include piecewise-defined functions, such as f(x) = {x^2 if x < 0, e^x if x ≥ 0}, and functions with removable discontinuities, such as f(x) = (x-1)/(x-1) for all x ≠ 1.

5. How can we use the epsilon-delta definition of continuity to find the continuity interval of an exponential function?

To find the continuity interval of an exponential function using the epsilon-delta definition, we must first find the limit of the function as x approaches a. Then, we can choose any epsilon value and use it to find the corresponding delta value. The continuity interval will be the range of x values where the function remains continuous.

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