If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Differential Calculus>Unit 1

Lesson 11: Continuity at a point

# Worked example: point where a function is continuous

In this video, we explore the limit of a piecewise function at the point where two cases of the function meet. By finding the left-hand and right-hand limits, we can determine if they're equal. If so, the limit exists at that point, and we've successfully analyzed the function's behavior.

## Want to join the conversation?

• at he substitutes x with 3 but the function is defined between 0 < x < 3. In other words it's defined up to three but not three. So isn't that technically wrong?
• in that particular function you mentioned,, it is continuos until x approaches 3.
We needed to find the limit of f(x) of that function as x approached 3. That is only the idea of limit.. It is not defined at x =3, but we can find a value of f(x) so that it would have formed a continuos graph.. So we find the value of f(x) that would have been for x=3.

Hope that was not complicated
• At how we can understand that function is continuous?
• It is not undefined for any positive argument. This means that there are no asymptotes or removable discontinuities, but proving continuity can be done in a variety of ways (for instance, noting that it is differentiable or noting that its inverse is differentiable etc.). Since differentiability is a stronger condition than continuity, all differentiable functions are also continuous over the differentiable interval.
• Can someone please suggest me a video in which all log functions are explained? Because I am only aware of the basic stuff but I guess as we proceed, we need to know the complicated functions of log too!
• I forgot what a log is :/
• A logarithm is essentially the opposite of the exponential function. What this means is that if a^x = b, the log(base a) b = x.
• why did he add 10 there suddenly??
• In math, "log" is short for "logarithm". It's a way to show how many times we need to multiply a number (the base) to get another number. When we say "log" without specifying a base, it's assumed to be base 10. That's a common math practice. So, he wasn't adding 10. He was explaining that "log" means "log base 10".

I recommend going back to Algebra 2 if you're still confused on logarithms: https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs
• What is a piecewise function?
• A piecewise function has different rules in different intervals. For example, look up aat this function:

f(x) = x^2 if x if x<4
= 4 if x<4 or x=4

Between the interval wich goes from negative infinity, it is x^2; and between the interval wich goes from 4 to positive infinity it is always four.

To give a counterexample, g(x)=x^2+1 is not a piecewise function, because it is always equal to x^2+1; without mattering the value of x
• At and , what if the function is non continuous? What numbers do we plug in for x?
• If there is a jump discontinuity, then the limit from the left side and the limit from the right side will not be equal so the overall limit does not exist. You still have to plug in the same x value in both equations and you will get different values so the overall limit does not exist. But if there is a removable discontinuity, both the limit from the left side and the limit from the right side will be equal so the overall limit exists. In any case, you have to plug in the same x value.
• for a limit to be continuous, lim(x tends to c) f(x) = f(c).

IN this case we know that lim(x tends to 3) g(x) = log(9) but we don't know if
g(3)=log(9).

So how can we say that the limit is continuous
• We do know that g(3)=log(9), because the function g is defined at x=3 and we can plug 3 into the function.

g(3) = (4-3)*log(9) = 1*log(9) = log(9)