积和商的连续性

Continuity of Products and Quotients

We all know what it means if a function is continuous- basically for any interval a to b, the function doesn't "skip" any points, and further for any c such that $a<c<b$: $\lim_{x \rightarrow c^{+}} f(x) = \lim_{x \rightarrow c^{-}} f(x)$.

This leads to the following idea: If f(x) and g(x) are continuous, then f(x)*g(x) is also continuous, but f(x)/g(x) may not be.

Allow me to provide a short proof.

Suppose f(x) and g(x) are continuous at some point q. Then:

$\lim_{x \rightarrow q} f(x) = f(q) = k$

$\lim_{x \rightarrow q} g(x) = g(q) = p$

Since both limits must approach a value by definition since they are continuous (if they didn't then the function could not be continuous.)

Then $\lim_{x \rightarrow q} f(x)*g(x) = f(q)*g(q) = k*p$

And $\lim_{x \rightarrow q} f(x)/g(x) = f(q)/g(q) = k/p$

k*p will clearly be a "safe" value for all q. However, k/p might not be defined! If g(x) as x approaches q equals zero then the funcion could be undefined!

Let me try to illustrate this point:

Take the functions f(x) and g(x) - both are continuous functions, clearly. As we showed above the product is then continuous, but the quotient is not! You have to be careful when dividing limits because it is possible to create a discontinuity.

To summarize the key points: -Multiplying two continuous limits together is always okay -Multiplying or dividing a function by a non-zero constant always preserves continuity -Dividing two continuous does not necessarily preserve continuity. Check to make sure the limit of the denominator is non-zero!