Question

Use the definition of continuity to find the constant such that the function is continuous
f(x) = x+3, x<-1
2x-c, x>-1

Answer 1

FOR A FUNCTION TO BE CONTINUOS, THE LIMIT AS x TENDS TO -1 FROM THE LEFT MUST BE EQUAL TO THE LIMIT AS x TENDS TO -1 FROM THE POSITIVE DIRECTION.

the limit when x<-1 is 2 and the limit when x>-1 is -2-c , the function can only be continuos if the two are equal so we equate them
2=-2-c
c=-4 the constant must be -4
so for the function to be continuos, we redefine it to look like
f(x)=x+3.x<-1
2x+4, x>-1

Answer 2

For a function (line of curve) to be continuous at a point, the limit to the point from both the sides have to be equal.
For example,
this is a continuous function


and this is not



Coming to your problem,

if we want your function to be continuous
X --> -1 from right has to be equal to X--> -1 from the left

X + 3 = 2X - C
Here, X= -1

There for
-1 + 3 = 2 (-1) - C
2 = -2 - C
Therefore,
C = -4

So, the function is redefined as
f(x)=x+3. x < -1
2x+4, x > -1

Answer 3

AbbasBagwala .fellow mathmatician your 1st curve looks like a REMOVABLE DISCONTINUITY CURVE