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Continuous Functions - Problem 1

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

One property of continuous functions is that the addition of two continuous functions is also continuous. For example, if you know that f(x) and g(x) are continuous, then you know that f(x) + g(x) is continuous. Additionally, scalar multiples (multiplying by a constant) of continuous functions are also continuous. So, if f(x) is continuous, then c*f(x) is continuous. From this, you can see that all polynomials are continuous, meaning that they are continuous at all points that they are defined.

I'm going to do a problem that involves proving that a function is continuous. First let's take a look at a rule. If two functions f, and g are continuous functions, and c is a constant. Then two things; first f(x) plus g(x) will be a continuous function. The constant c times f(x) will also be continuous. So these functions will be continuous wherever they are defined.

Let's take a look at the problem. Explain why the polynomial p(x) equals 3x² minus x plus 10 is a continuous function. First of all, the function is x², x, and 10, these guys, those are power functions. Therefore they are continuous. So are continuous, because they are power functions.

Second, when you multiply a continuous function by a constant, it remains continuous. So 3 times x², and -1 times x are continuous functions. So 3 times x², and -x are continuous. Why would that be? This would be by property 2 here.

Finally, if you add these up, they're continuous by property 1. P(x) which equals 3x² minus x plus 10 is continuous by property 1. Sums, and it turns also differences of continuous functions are continuous. Now using an argument like this, you can actually show that any polynomial functions are continuous. That's my next result.

All polynomials are continuous. Remember continuous means that they're continuous wherever they're defined. Polynomial functions are always defined for all real numbers. So polynomials are always continuous for all real numbers. Here is one example.

Another example would be say q(x) equals 0.08x³ plus 12x. R(x) equals 4x to the 12th plus 8x² plus 100. All three of these are polynomial functions. Therefore, they're all continuous for all real numbers.

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