Find maximum value of a continuous function at a specific resolution

Find maximum value of a continuous function at a specific resolution

Imagine having a function that is continuous over a range [0.0,n]. Are
there any algorithms to find the maximum value of the function given a
minimum step size s more quickly than simple iteration? The simple
iteration is straightforward to program but the time complexity grows when
n / s is large.
double maxValue = 0;
double maxValueX = 0;
double s = 0.1 * n;
for (double x = 0.0; x <= n; x += s)
{
double value = someFunction(x);
if(value > maxValue) {
maxValue = value;
maxValueX = x;
}
}
I have tried this approach which is much quicker, but don't know if it
will get stuck on local maximums.
double min = 0;
double max = n;
int steps = 10;
increment = (max - min) / steps;
while (increment > s)
{
double maxValue = 0;
double maxValueX = X;
for (double x= min; x <= max; x+= increment)
{
double value = someFunction(x);
if(value > maxValue) {
maxValue = value;
maxValueX = x;
}
}
min = Math.Max(maxValueX - increment, 0.0);
max = Math.Min(maxValueX + increment, n);
increment = (max - min) / steps;
}