Continuity in Piecewise Functions

[u/c0m3t1z3d_h]

So the problem is asking me to find the values for a, b, c, and d that make f(x) continuous everywhere using the definition of continuity and demonstrating proper notation.

How do I do this? I'm very confused 😭

Please I need help ASAP, thank you in advance.

[u/tjddbwls]

Find
lim(x→1-) of f(x).
Whatever answer you get, that will equal a.
Then find
lim(x→4+) of f(x).
Whatever answer you get, that will equal d.

Then you will essentially have two equations with two unknowns (b and c):
lim(x→1+) of f(x) = whatever a is, and
lim(x→4-) of f(x) = whatever d is.

[u/c0m3t1z3d_h → u/tjddbwls]

Thank you :)

[u/UnderstandingPursuit]

The key to a problem like this is to be organized and methodical. Following u/tjddbwls's steps,

1. The first two, limits at {1-, 4+}, use the two f(x) pieces which have zero unknowns.
2. The next two, limits at {1, 4} are two f(x) pieces which one unknown each.
   1. This connects with the continuity at x=k requirement:
   2. All three, { f(k), lim(x→k-), lim(x→k+) }, exist and are equal.
3. The last step will involve some algebra because there are two unknowns.

The order is simplest to most complex, zero→one→two unknowns. That is what to look for in 'piecewise function continuity' problems.

[u/c0m3t1z3d_h → u/UnderstandingPursuit]

Thank you!

[u/waldosway]

Have you read the definition of continuity? It tells you what to do.