Question

Help me! For which values ​​of a is f (x) continuous for all x??

Answer 1

For `x < pi`, `f(x) = asinx` is a constant times the sine function, which is continuous at `x`.
For `x > pi`, `f(x) = x+a` is a linear function which is continuous at `x`.

In order for `f` to be continuous at `pi`, we need

`lim_(xrarrpi)f(x) = f(pi)`

Since the rule changes at `pi`, we'll need to look at the left and right limit and make sure they are the same.

`lim_(xrarrpi^-)f(x) = lim_(xrarrpi^-)asinx = asinpi = a(0) = 0` regardless of the value of `a`.

`lim_(xrarrpi^+)f(x) = lim_(xrarrpi^+)(x+ a) = pi+a`

In order for the two one-sided limits to be equal, we must have `a = -pi`

Finally, we check to be sure that `f(pi) = 0`. (It does,)